Singular value decomposition

In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix into a rotation, followed by a rescaling followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix. It is related to the polar decomposition.

Specifically, the singular value decomposition of an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} complex matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is a factorization of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \mathbf{U\Sigma V^*},} where ⁠ $\mathbf {U}$ ⁠ is an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times m} ⁠ complex unitary matrix, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} rectangular diagonal matrix with non-negative real numbers on the diagonal, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} complex unitary matrix, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} is the conjugate transpose of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠. Such decomposition always exists for any complex matrix. If ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is real, then ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U \mathbf \Sigma \mathbf V^\mathrm{T}.}

The diagonal entries Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i = \Sigma_{i i}} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} are uniquely determined by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ and are known as the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠. The number of non-zero singular values is equal to the rank of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠. The columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and the columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are called left-singular vectors and right-singular vectors of ⁠ $\mathbf {M}$ ⁠, respectively. They form two sets of orthonormal bases ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1, \ldots, \mathbf u_m} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v_1, \ldots, \mathbf v_n,} ⁠ and if they are sorted so that the singular values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \sum_{i=1}^{r}\sigma_i\mathbf{u}_i\mathbf{v}_i^{*}, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \leq \min\{m,n\}} is the rank of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠

The SVD is not unique, however it is always possible to choose the decomposition such that the singular values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Sigma_{i i}} are in descending order. In this case, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} (but not ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠) is uniquely determined by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠

The term sometimes refers to the compact SVD, a similar decomposition ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf{U\Sigma V}^*} ⁠ in which ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ is square diagonal of size ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \times r,} ⁠ where ⁠ $r\leq \min\{m,n\}$ ⁠ is the rank of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ and has only the non-zero singular values. In this variant, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ is an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times r} ⁠ semi-unitary matrix and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} is an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times r} ⁠ semi-unitary matrix, such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U^* \mathbf U = \mathbf V^* \mathbf V = \mathbf I_r.}

Mathematical applications of the SVD include computing the pseudoinverse, matrix approximation, and determining the rank, range, and null space of a matrix. The SVD is also extremely useful in all areas of science, engineering, and statistics, such as signal processing, least squares fitting of data, and process control.

Intuitive interpretations

Rotation, coordinate scaling, and reflection

In the special case when ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times m} ⁠ real square matrix, the matrices ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠ $\mathbf {V} ^{*}$ ⁠ can be chosen to be real ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times m} ⁠ matrices too. In that case, "unitary" is the same as "orthogonal". Then, interpreting both unitary matrices as well as the diagonal matrix, summarized here as ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A,} ⁠ as a linear transformation ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x \mapsto \mathbf{Ax} } ⁠ of the space ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R_m,} ⁠ the matrices ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ represent rotations or reflection of the space, while ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ represents the scaling of each coordinate ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x_i} ⁠ by the factor ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i.} ⁠ Thus the SVD decomposition breaks down any linear transformation of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^m} ⁠ into a composition of three geometrical transformations: a rotation or reflection (⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠), followed by a coordinate-by-coordinate scaling (⁠ $\mathbf {\Sigma }$ ⁠), followed by another rotation or reflection (⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠).

In particular, if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ has a positive determinant, then ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ can be chosen to be both rotations with reflections, or both rotations without reflections.^{[citation needed]} If the determinant is negative, exactly one of them will have a reflection. If the determinant is zero, each can be independently chosen to be of either type.

If the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is real but not square, namely ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\times n} ⁠ with ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \neq n,} ⁠ it can be interpreted as a linear transformation from ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^n} ⁠ to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^ m.} ⁠ Then ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ can be chosen to be rotations/reflections of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^m} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^n,} ⁠ respectively; and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma,} ⁠ besides scaling the first ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \min\{m,n\} } ⁠ coordinates, also extends the vector with zeros, i.e. removes trailing coordinates, so as to turn ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^n} ⁠ into ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^m.} ⁠

Singular values as semiaxes of an ellipse or ellipsoid

As shown in the figure, the singular values can be interpreted as the magnitude of the semiaxes of an ellipse in 2D. This concept can be generalized to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ⁠-dimensional Euclidean space, with the singular values of any ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} ⁠ square matrix being viewed as the magnitude of the semiaxis of an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ⁠-dimensional ellipsoid. Similarly, the singular values of any ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix can be viewed as the magnitude of the semiaxis of an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ⁠-dimensional ellipsoid in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} ⁠-dimensional space, for example as an ellipse in a (tilted) 2D plane in a 3D space. Singular values encode magnitude of the semiaxis, while singular vectors encode direction. See below for further details.

The columns of $U$ and $V$ are orthonormal bases

Since ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ are unitary, the columns of each of them form a set of orthonormal vectors, which can be regarded as basis vectors. The matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ maps the basis vector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_i} ⁠ to the stretched unit vector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i \mathbf U_i.} ⁠ By the definition of a unitary matrix, the same is true for their conjugate transposes ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U^*} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V,} ⁠ except the geometric interpretation of the singular values as stretches is lost. In short, the columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U,} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U^*,} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V,} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ are orthonormal bases. When ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is a positive-semidefinite Hermitian matrix, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are both equal to the unitary matrix used to diagonalize ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ However, when ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is not positive-semidefinite and Hermitian but still diagonalizable, its eigendecomposition and singular value decomposition are distinct.

Relation to the four fundamental subspaces

The first ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠ columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ are a basis of the column space of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠.
The last ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m-r} ⁠ columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ are a basis of the null space of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^*} ⁠.
The first ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠ columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are a basis of the column space of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^*} ⁠ (the row space of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ in the real case).
The last ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n-r} ⁠ columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are a basis of the null space of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠.

Geometric meaning

Because ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are unitary, we know that the columns ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_1, \ldots, \mathbf U_m} ⁠ of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ yield an orthonormal basis of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^m} ⁠ and the columns ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_1, \ldots, \mathbf V_n} ⁠ of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ yield an orthonormal basis of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^n} ⁠ (with respect to the standard scalar products on these spaces).

The linear transformation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T : \left\{\begin{aligned} K^n &\to K^m \\ x &\mapsto \mathbf{M}x \end{aligned}\right. }

has a particularly simple description with respect to these orthonormal bases: we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathbf{V}_i) = \sigma_i \mathbf{U}_i, \qquad i = 1, \ldots, \min(m, n), }

where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ is the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} ⁠-th diagonal entry of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma,} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(\mathbf V_i) = 0} ⁠ for ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i > \min(m,n).} ⁠

The geometric content of the SVD theorem can thus be summarized as follows: for every linear map ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T : K^n \to K^m } ⁠ one can find orthonormal bases of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^n} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^m} ⁠ such that ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ⁠ maps the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} ⁠-th basis vector of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^n} ⁠ to a non-negative multiple of the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} ⁠-th basis vector of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^m,} ⁠ and sends the leftover basis vectors to zero. With respect to these bases, the map ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ⁠ is therefore represented by a diagonal matrix with non-negative real diagonal entries.

To get a more visual flavor of singular values and SVD factorization – at least when working on real vector spaces – consider the sphere ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} ⁠ of radius one in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^n.} ⁠ The linear map ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ⁠ maps this sphere onto an ellipsoid in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^m.} ⁠ Non-zero singular values are simply the lengths of the semi-axes of this ellipsoid. Especially when ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = m,} ⁠ and all the singular values are distinct and non-zero, the SVD of the linear map ⁠ $T$ ⁠ can be easily analyzed as a succession of three consecutive moves: consider the ellipsoid ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S)} ⁠ and specifically its axes; then consider the directions in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^n} ⁠ sent by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ⁠ onto these axes. These directions happen to be mutually orthogonal. Apply first an isometry ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ sending these directions to the coordinate axes of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R^n.} ⁠ On a second move, apply an endomorphism ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D} ⁠ diagonalized along the coordinate axes and stretching or shrinking in each direction, using the semi-axes lengths of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S)} ⁠ as stretching coefficients. The composition ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D \circ \mathbf V^*} ⁠ then sends the unit-sphere onto an ellipsoid isometric to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S).} ⁠ To define the third and last move, apply an isometry ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ to this ellipsoid to obtain ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T(S).} ⁠ As can be easily checked, the composition ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U \circ \mathbf D \circ \mathbf V^*} ⁠ coincides with ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T.} ⁠

Example

Consider the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4 \times 5} ⁠ matrix

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \begin{bmatrix} 1 & 0 & 0 & 0 & 2 \\ 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 & 0 \end{bmatrix} }

A singular value decomposition of this matrix is given by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U\mathbf \Sigma \mathbf V^*} ⁠

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{U} &= \begin{bmatrix} \color{Green}0 & \color{Blue}-1 & \color{Cyan}0 & \color{Emerald}0 \\ \color{Green}-1 & \color{Blue}0 & \color{Cyan}0 & \color{Emerald}0 \\ \color{Green}0 & \color{Blue}0 & \color{Cyan}0 & \color{Emerald}-1 \\ \color{Green}0 & \color{Blue}0 & \color{Cyan}-1 & \color{Emerald}0 \end{bmatrix} \\[6pt] \mathbf \Sigma &= \begin{bmatrix} 3 & 0 & 0 & 0 & \color{Gray}\mathit{0} \\ 0 & \sqrt{5} & 0 & 0 & \color{Gray}\mathit{0} \\ 0 & 0 & 2 & 0 & \color{Gray}\mathit{0} \\ 0 & 0 & 0 & \color{Red}\mathbf{0} & \color{Gray}\mathit{0} \end{bmatrix} \\[6pt] \mathbf{V}^* &= \begin{bmatrix} \color{Violet}0 & \color{Violet}0 & \color{Violet}-1 & \color{Violet}0 &\color{Violet}0 \\ \color{Plum}-\sqrt{0.2}& \color{Plum}0 & \color{Plum}0 & \color{Plum}0 &\color{Plum}-\sqrt{0.8} \\ \color{Magenta}0 & \color{Magenta}-1 & \color{Magenta}0 & \color{Magenta}0 &\color{Magenta}0 \\ \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}1 &\color{Orchid}0 \\ \color{Purple} - \sqrt{0.8} & \color{Purple}0 & \color{Purple}0 & \color{Purple}0 & \color{Purple}\sqrt{0.2} \end{bmatrix} \end{align}}

The scaling matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ is zero outside of the diagonal (grey italics) and one diagonal element is zero (red bold, light blue bold in dark mode). Furthermore, because the matrices ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ are unitary, multiplying by their respective conjugate transposes yields identity matrices, as shown below. In this case, because ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ are real valued, each is an orthogonal matrix.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{U} \mathbf{U}^* &= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} = \mathbf{I}_4 \\[6pt] \mathbf{V} \mathbf{V}^* &= \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{bmatrix} = \mathbf{I}_5 \end{align}}

This particular singular value decomposition is not unique. Choosing ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}^* = \begin{bmatrix} \color{Violet}0 & \color{Violet}1 & \color{Violet}0 & \color{Violet}0 & \color{Violet}0 \\ \color{Plum}0 & \color{Plum}0 & \color{Plum}1 & \color{Plum}0 & \color{Plum}0 \\ \color{Magenta}\sqrt{0.2} & \color{Magenta}0 & \color{Magenta}0 & \color{Magenta}0 & \color{Magenta}\sqrt{0.8} \\ \color{Orchid}\sqrt{0.4} & \color{Orchid}0 & \color{Orchid}0 & \color{Orchid}\sqrt{0.5} & \color{Orchid}-\sqrt{0.1} \\ \color{Purple}-\sqrt{0.4} & \color{Purple}0 & \color{Purple}0 & \color{Purple}\sqrt{0.5} & \color{Purple}\sqrt{0.1} \end{bmatrix}}

is also a valid singular value decomposition.

SVD and spectral decomposition

Singular values, singular vectors, and their relation to the SVD

A non-negative real number ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} ⁠ is a singular value for ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ if and only if there exist unit-length vectors ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^m} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v} ⁠ in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^n} ⁠ such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{M v} &= \sigma \mathbf{u}, \\[3mu] \mathbf M^*\mathbf u &= \sigma \mathbf{v}. \end{align}}

The vectors ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v} ⁠ are called left-singular and right-singular vectors for ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma,} ⁠ respectively.

In any singular value decomposition

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^* }

the diagonal entries of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ are equal to the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ The first ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \min(m,n)} ⁠ columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are, respectively, left- and right-singular vectors for the corresponding singular values. Consequently, the above theorem implies that:

An ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix ⁠ $\mathbf {M}$ ⁠ has at most ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ⁠ distinct singular values.
It is always possible to find a unitary basis ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ for ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^m} ⁠ with a subset of basis vectors spanning the left-singular vectors of each singular value of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠
It is always possible to find a unitary basis ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ for ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^n} ⁠ with a subset of basis vectors spanning the right-singular vectors of each singular value of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠

A singular value for which we can find two left (or right) singular vectors that are linearly independent is called degenerate. If ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_2} ⁠ are two left-singular vectors which both correspond to the singular value σ, then any normalized linear combination of the two vectors is also a left-singular vector corresponding to the singular value σ. The similar statement is true for right-singular vectors. The number of independent left and right-singular vectors coincides, and these singular vectors appear in the same columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ corresponding to diagonal elements of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ all with the same value ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma.} ⁠

As an exception, the left and right-singular vectors of singular value 0 comprise all unit vectors in the cokernel and kernel, respectively, of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ which by the rank–nullity theorem cannot be the same dimension if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \neq n.} ⁠ Even if all singular values are nonzero, if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m > n} ⁠ then the cokernel is nontrivial, in which case ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ is padded with ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m - n} ⁠ orthogonal vectors from the cokernel. Conversely, if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m < n,} ⁠ then ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ is padded by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n - m} ⁠ orthogonal vectors from the kernel. However, if the singular value of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} ⁠ exists, the extra columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ or ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ already appear as left or right-singular vectors.

Non-degenerate singular values always have unique left- and right-singular vectors, up to multiplication by a unit-phase factor ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\varphi} } ⁠ (for the real case up to a sign). Consequently, if all singular values of a square matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ are non-degenerate and non-zero, then its singular value decomposition is unique, up to multiplication of a column of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ by a unit-phase factor and simultaneous multiplication of the corresponding column of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ by the same unit-phase factor. In general, the SVD is unique up to arbitrary unitary transformations applied uniformly to the column vectors of both ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ spanning the subspaces of each singular value, and up to arbitrary unitary transformations on vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ spanning the kernel and cokernel, respectively, of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠

Relation to eigenvalue decomposition

The singular value decomposition is very general in the sense that it can be applied to any ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix, whereas eigenvalue decomposition can only be applied to square diagonalizable matrices. Nevertheless, the two decompositions are related.

If ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ has SVD ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^*,} ⁠ the following two relations hold:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{M}^* \mathbf{M} &= \mathbf{V} \mathbf \Sigma^* \mathbf{U}^*\, \mathbf{U} \mathbf \Sigma \mathbf{V}^* = \mathbf{V} (\mathbf \Sigma^* \mathbf \Sigma) \mathbf{V}^*, \\[3mu] \mathbf{M} \mathbf{M}^* &= \mathbf{U} \mathbf \Sigma \mathbf{V}^*\, \mathbf{V} \mathbf \Sigma^* \mathbf{U}^* = \mathbf{U} (\mathbf \Sigma \mathbf \Sigma^*) \mathbf{U}^*. \end{align}}

The right-hand sides of these relations describe the eigenvalue decompositions of the left-hand sides. Consequently:

The columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ (referred to as right-singular vectors) are eigenvectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M.} ⁠
The columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ (referred to as left-singular vectors) are eigenvectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*.} ⁠
The non-zero elements of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ (non-zero singular values) are the square roots of the non-zero eigenvalues of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠ or ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*.} ⁠

In the special case of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ being a normal matrix, and thus also square, the spectral theorem ensures that it can be unitarily diagonalized using a basis of eigenvectors, and thus decomposed as ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U\mathbf D\mathbf U^*} ⁠ for some unitary matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and diagonal matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D} ⁠ with complex elements ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ along the diagonal. When ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is positive semi-definite, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ will be non-negative real numbers so that the decomposition ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U \mathbf D \mathbf U^*} ⁠ is also a singular value decomposition. Otherwise, it can be recast as an SVD by moving the phase ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\varphi} } ⁠ of each ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ to either its corresponding ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_i} ⁠ or ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_i.} ⁠ The natural connection of the SVD to non-normal matrices is through the polar decomposition theorem: ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf S \mathbf R,} ⁠ where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf S = \mathbf U \mathbf\Sigma \mathbf U^*} ⁠ is positive semidefinite and normal, and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R = \mathbf U \mathbf V^*} ⁠ is unitary.

Thus, except for positive semi-definite matrices, the eigenvalue decomposition and SVD of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ while related, differ: the eigenvalue decomposition is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{{1}}}} ⁠ where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ is not necessarily unitary and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D} ⁠ is not necessarily positive semi-definite, while the SVD is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{{1}}}} ⁠ where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ⁠ is diagonal and positive semi-definite, and ⁠ $\mathbf {U}$ ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ are unitary matrices that are not necessarily related except through the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ While only non-defective square matrices have an eigenvalue decomposition, any ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix has a SVD.

Applications of the SVD

Pseudoinverse

The singular value decomposition can be used for computing the pseudoinverse of a matrix. The pseudoinverse of the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ with singular value decomposition ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^*} ⁠ is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^+ = \mathbf V \boldsymbol \Sigma^+ \mathbf U^\ast, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol \Sigma^+} is the pseudoinverse of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \boldsymbol \Sigma} , which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix. The pseudoinverse is one way to solve linear least squares problems.

Solving homogeneous linear equations

A set of homogeneous linear equations can be written as ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A \mathbf x = \mathbf 0} ⁠ for a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ and vector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x.} ⁠ A typical situation is that ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ is known and a non-zero ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠ is to be determined which satisfies the equation. Such an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠ belongs to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠'s null space and is sometimes called a (right) null vector of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ The vector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠ can be characterized as a right-singular vector corresponding to a singular value of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ that is zero. This observation means that if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ is a square matrix and has no vanishing singular value, the equation has no non-zero ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠ as a solution. It also means that if there are several vanishing singular values, any linear combination of the corresponding right-singular vectors is a valid solution. Analogously to the definition of a (right) null vector, a non-zero ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠ satisfying ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x^* \mathbf A = \mathbf 0} ⁠ with ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x^*} ⁠ denoting the conjugate transpose of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x,} ⁠ is called a left null vector of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠

Total least squares minimization

A total least squares problem seeks the vector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠ that minimizes the 2-norm of a vector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A \mathbf x} ⁠ under the constraint Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \mathbf x \| = 1.} The solution turns out to be the right-singular vector of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ corresponding to the smallest singular value.

Range, null space and rank

Another application of the SVD is that it provides an explicit representation of the range and null space of a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ The right-singular vectors corresponding to vanishing singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ span the null space of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ and the left-singular vectors corresponding to the non-zero singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ span the range of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ For example, in the above example the null space is spanned by the last row of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ and the range is spanned by the first three columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U.} ⁠

As a consequence, the rank of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ equals the number of non-zero singular values which is the same as the number of non-zero diagonal elements in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} . In numerical linear algebra the singular values can be used to determine the effective rank of a matrix, as rounding error may lead to small but non-zero singular values in a rank deficient matrix. Singular values beyond a significant gap are assumed to be numerically equivalent to zero.

Low-rank matrix approximation

Some practical applications need to solve the problem of approximating a matrix ⁠ $\mathbf {M}$ ⁠ with another matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\mathbf{M}}} , said to be truncated, which has a specific rank ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠. In the case that the approximation is based on minimizing the Frobenius norm of the difference between ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\mathbf M} } ⁠ under the constraint that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{rank}\bigl(\tilde{\mathbf{M}}\bigr) = r,} it turns out that the solution is given by the SVD of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ namely

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\mathbf{M}} = \mathbf{U} \tilde{\mathbf \Sigma} \mathbf{V}^*, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\mathbf \Sigma}} is the same matrix as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} except that it contains only the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠ largest singular values (the other singular values are replaced by zero). This is known as the Eckart–Young theorem, as it was proved by those two authors in 1936 (although it was later found to have been known to earlier authors; see Stewart 1993).

Separable models

The SVD can be thought of as decomposing a matrix into a weighted, ordered sum of separable matrices. By separable, we mean that a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ can be written as an outer product of two vectors ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A = \mathbf u \otimes \mathbf v,} ⁠ or, in coordinates, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ij} = u_i v_j.} ⁠ Specifically, the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ can be decomposed as,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \sum_i \mathbf{A}_i = \sum_i \sigma_i \mathbf U_i \otimes \mathbf V_i. }

Here ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_i} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_i} ⁠ are the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} ⁠-th columns of the corresponding SVD matrices, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ are the ordered singular values, and each ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A_i} ⁠ is separable. The SVD can be used to find the decomposition of an image processing filter into separable horizontal and vertical filters. Note that the number of non-zero ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ is exactly the rank of the matrix.^{[citation needed]} Separable models often arise in biological systems, and the SVD factorization is useful to analyze such systems. For example, some visual area V1 simple cells' receptive fields can be well described^[1] by a Gabor filter in the space domain multiplied by a modulation function in the time domain. Thus, given a linear filter evaluated through, for example, reverse correlation, one can rearrange the two spatial dimensions into one dimension, thus yielding a two-dimensional filter (space, time) which can be decomposed through SVD. The first column of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ in the SVD factorization is then a Gabor while the first column of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ represents the time modulation (or vice versa). One may then define an index of separability

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{\sigma_1^2}{\sum_i \sigma_i^2}, }

which is the fraction of the power in the matrix M which is accounted for by the first separable matrix in the decomposition.^[2]

Nearest orthogonal matrix

It is possible to use the SVD of a square matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ to determine the orthogonal matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf O} ⁠ closest to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ The closeness of fit is measured by the Frobenius norm of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf O - \mathbf A.} ⁠ The solution is the product ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U \mathbf V^*.} ⁠^[3] This intuitively makes sense because an orthogonal matrix would have the decomposition ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U \mathbf I \mathbf V^*} ⁠ where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf I} ⁠ is the identity matrix, so that if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A = \mathbf U \mathbf \Sigma \mathbf V^*} ⁠ then the product ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A = \mathbf U \mathbf V^*} ⁠ amounts to replacing the singular values with ones. Equivalently, the solution is the unitary matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R = \mathbf U \mathbf V^*} ⁠ of the Polar Decomposition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf R \mathbf P = \mathbf P' \mathbf R} in either order of stretch and rotation, as described above.

A similar problem, with interesting applications in shape analysis, is the orthogonal Procrustes problem, which consists of finding an orthogonal matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf O} ⁠ which most closely maps ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf B.} ⁠ Specifically,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{O} = \underset\Omega\operatorname{argmin} \|\mathbf{A}\boldsymbol{\Omega} - \mathbf{B}\|_F \quad\text{subject to}\quad \boldsymbol{\Omega}^\operatorname{T}\boldsymbol{\Omega} = \mathbf{I}, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \cdot \|_F} denotes the Frobenius norm.

This problem is equivalent to finding the nearest orthogonal matrix to a given matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf A^\operatorname{T} \mathbf B} .

The Kabsch algorithm

The Kabsch algorithm (called Wahba's problem in other fields) uses SVD to compute the optimal rotation (with respect to least-squares minimization) that will align a set of points with a corresponding set of points. It is used, among other applications, to compare the structures of molecules.

Signal processing

The SVD and pseudoinverse have been successfully applied to signal processing,^[4] image processing^[5] and big data (e.g., in genomic signal processing).^[6]^[7]^[8]^[9]

Other examples

The SVD is also applied extensively to the study of linear inverse problems and is useful in the analysis of regularization methods such as that of Tikhonov. It is widely used in statistics, where it is related to principal component analysis and to correspondence analysis, and in signal processing and pattern recognition. It is also used in output-only modal analysis, where the non-scaled mode shapes can be determined from the singular vectors. Yet another usage is latent semantic indexing in natural-language text processing.

In general numerical computation involving linear or linearized systems, there is a universal constant that characterizes the regularity or singularity of a problem, which is the system's "condition number" Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa := \sigma_\text{max} / \sigma_\text{min}} . It often controls the error rate or convergence rate of a given computational scheme on such systems.^[10]^[11]

The SVD also plays a crucial role in the field of quantum information, in a form often referred to as the Schmidt decomposition. Through it, states of two quantum systems are naturally decomposed, providing a necessary and sufficient condition for them to be entangled: if the rank of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} matrix is larger than one.

One application of SVD to rather large matrices is in numerical weather prediction, where Lanczos methods are used to estimate the most linearly quickly growing few perturbations to the central numerical weather prediction over a given initial forward time period; i.e., the singular vectors corresponding to the largest singular values of the linearized propagator for the global weather over that time interval. The output singular vectors in this case are entire weather systems. These perturbations are then run through the full nonlinear model to generate an ensemble forecast, giving a handle on some of the uncertainty that should be allowed for around the current central prediction.

SVD has also been applied to reduced order modelling. The aim of reduced order modelling is to reduce the number of degrees of freedom in a complex system which is to be modeled. SVD was coupled with radial basis functions to interpolate solutions to three-dimensional unsteady flow problems.^[12]

Interestingly, SVD has been used to improve gravitational waveform modeling by the ground-based gravitational-wave interferometer aLIGO.^[13] SVD can help to increase the accuracy and speed of waveform generation to support gravitational-waves searches and update two different waveform models.

Singular value decomposition is used in recommender systems to predict people's item ratings.^[14] Distributed algorithms have been developed for the purpose of calculating the SVD on clusters of commodity machines.^[15]

Low-rank SVD has been applied for hotspot detection from spatiotemporal data with application to disease outbreak detection.^[16] A combination of SVD and higher-order SVD also has been applied for real time event detection from complex data streams (multivariate data with space and time dimensions) in disease surveillance.^[17]

In astrodynamics, the SVD and its variants are used as an option to determine suitable maneuver directions for transfer trajectory design^[18] and orbital station-keeping.^[19]

Proof of existence

An eigenvalue ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} ⁠ of a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is characterized by the algebraic relation ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf u = \lambda \mathbf u.} ⁠ When ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is Hermitian, a variational characterization is also available. Let ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ be a real ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} ⁠ symmetric matrix. Define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \left\{ \begin{align} \R^n &\to \R \\ \mathbf{x} &\mapsto \mathbf{x}^\operatorname{T} \mathbf{M} \mathbf{x} \end{align}\right.}

By the extreme value theorem, this continuous function attains a maximum at some ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ when restricted to the unit sphere Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\|\mathbf x\| = 1\}.} By the Lagrange multipliers theorem, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ necessarily satisfies

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{u} - \lambda \cdot \nabla \mathbf{u}^\operatorname{T} \mathbf{u} = 0}

for some real number ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda.} ⁠ The nabla symbol, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla} ⁠, is the del operator (differentiation with respect to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} ⁠). Using the symmetry of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \mathbf{x}^\operatorname{T} \mathbf{M} \mathbf{x} - \lambda \cdot \nabla \mathbf{x}^\operatorname{T} \mathbf{x} = 2(\mathbf{M}-\lambda \mathbf{I})\mathbf{x}.}

Therefore ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf u = \lambda \mathbf u,} ⁠ so ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ is a unit length eigenvector of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ For every unit length eigenvector ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v} ⁠ of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ its eigenvalue is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\mathbf v),} ⁠ so ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} ⁠ is the largest eigenvalue of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ The same calculation performed on the orthogonal complement of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ gives the next largest eigenvalue and so on. The complex Hermitian case is similar; there ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(\mathbf x) = \mathbf x^* \mathbf M \mathbf x} ⁠ is a real-valued function of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} ⁠ real variables.

Singular values are similar in that they can be described algebraically or from variational principles. Although, unlike the eigenvalue case, Hermiticity, or symmetry, of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is no longer required.

This section gives these two arguments for existence of singular value decomposition.

Based on the spectral theorem

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}} be an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ complex matrix. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} is positive semi-definite and Hermitian, by the spectral theorem, there exists an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} ⁠ unitary matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^* \mathbf M^* \mathbf M \mathbf V = \bar\mathbf{D} = \begin{bmatrix} \mathbf{D} & 0 \\ 0 & 0\end{bmatrix}, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D}} is diagonal and positive definite, of dimension $\ell \times \ell$ , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} the number of non-zero eigenvalues of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} (which can be shown to verify Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell\le\min(n,m)} ). Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} is here by definition a matrix whose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} -th column is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} -th eigenvector of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} , corresponding to the eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\mathbf{D}}_{ii}} . Moreover, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} -th column of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} , for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j>\ell} , is an eigenvector of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} with eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\mathbf{D}}_{jj}=0} . This can be expressed by writing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}=\begin{bmatrix}\mathbf{V}_1 &\mathbf{V}_2\end{bmatrix}} , where the columns of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_2} therefore contain the eigenvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} corresponding to non-zero and zero eigenvalues, respectively. Using this rewriting of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} , the equation becomes:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \mathbf{V}_1^* \\ \mathbf{V}_2^* \end{bmatrix} \mathbf{M}^* \mathbf{M}\, \begin{bmatrix} \mathbf{V}_1 & \!\! \mathbf{V}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{V}_1^* \mathbf{M}^* \mathbf{M} \mathbf{V}_1 & \mathbf{V}_1^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2 \\ \mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_1 & \mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{D} & 0 \\ 0 & 0 \end{bmatrix}.}

This implies that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_1^* \mathbf{M}^* \mathbf{M} \mathbf{V}_1 = \mathbf{D}, \quad \mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2 = \mathbf{0}. }

Moreover, the second equation implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}\mathbf{V}_2 = \mathbf{0}} .^[20] Finally, the unitary-ness of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}} translates, in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_2} , into the following conditions:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{V}_1^* \mathbf{V}_1 &= \mathbf{I}_1, \\ \mathbf{V}_2^* \mathbf{V}_2 &= \mathbf{I}_2, \\ \mathbf{V}_1 \mathbf{V}_1^* + \mathbf{V}_2 \mathbf{V}_2^* &= \mathbf{I}_{12}, \end{align}}

where the subscripts on the identity matrices are used to remark that they are of different dimensions.

Let us now define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1 = \mathbf{M} \mathbf{V}_1 \mathbf{D}^{-\frac{1}{2}}. }

Then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1 \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* = \mathbf{M} \mathbf{V}_1 \mathbf{D}^{-\frac{1}{2}} \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* = \mathbf{M} (\mathbf{I} - \mathbf{V}_2\mathbf{V}_2^*) = \mathbf{M} - (\mathbf{M}\mathbf{V}_2)\mathbf{V}_2^* = \mathbf{M}, }

since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}\mathbf{V}_2 = \mathbf{0}. } This can be also seen as immediate consequence of the fact that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}\mathbf{V}_1\mathbf{V}_1^* = \mathbf{M}} . This is equivalent to the observation that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\boldsymbol v_i\}_{i=1}^\ell} is the set of eigenvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} corresponding to non-vanishing eigenvalues Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\lambda_i\}_{i=1}^\ell} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf M \boldsymbol v_i\}_{i=1}^\ell} is a set of orthogonal vectors, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl\{\lambda_i^{-1/2}\mathbf M \boldsymbol v_i\bigr\}\vphantom|_{i=1}^\ell} is a (generally not complete) set of orthonormal vectors. This matches with the matrix formalism used above denoting with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_1} the matrix whose columns are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\boldsymbol v_i\}_{i=1}^\ell} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_2} the matrix whose columns are the eigenvectors of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M}^* \mathbf{M}} with vanishing eigenvalue, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1} the matrix whose columns are the vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigl\{\lambda_i^{-1/2}\mathbf M \boldsymbol v_i\bigr\}\vphantom|_{i=1}^\ell} .

We see that this is almost the desired result, except that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{V}_1} are in general not unitary, since they might not be square. However, we do know that the number of rows of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1} is no smaller than the number of columns, since the dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{D}} is no greater than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . Also, since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1^*\mathbf{U}_1 = \mathbf{D}^{-\frac{1}{2}}\mathbf{V}_1^*\mathbf{M}^*\mathbf{M} \mathbf{V}_1 \mathbf{D}^{-\frac{1}{2}}=\mathbf{D}^{-\frac{1}{2}}\mathbf{D}\mathbf{D}^{-\frac{1}{2}} = \mathbf{I_1}, }

the columns in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_1} are orthonormal and can be extended to an orthonormal basis. This means that we can choose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U}_2} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{U} = \begin{bmatrix} \mathbf{U}_1 & \mathbf{U}_2 \end{bmatrix}} is unitary.

For ⁠ $\mathbf {V} _{1}$ ⁠ we already have ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_2} ⁠ to make it unitary. Now, define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma = \begin{bmatrix} \begin{bmatrix} \mathbf{D}^\frac{1}{2} & 0 \\ 0 & 0 \end{bmatrix} \\ 0 \end{bmatrix}, }

where extra zero rows are added or removed to make the number of zero rows equal the number of columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_2,} ⁠ and hence the overall dimensions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} equal to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\times n} . Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \mathbf{U}_1 & \mathbf{U}_2 \end{bmatrix} \begin{bmatrix} \begin{bmatrix} \mathbf{}D^\frac{1}{2} & 0 \\ 0 & 0 \end{bmatrix} \\ 0 \end{bmatrix} \begin{bmatrix} \mathbf{V}_1 & \mathbf{V}_2 \end{bmatrix}^* = \begin{bmatrix} \mathbf{U}_1 & \mathbf{U}_2 \end{bmatrix} \begin{bmatrix} \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* \\ 0 \end{bmatrix} = \mathbf{U}_1 \mathbf{D}^\frac{1}{2} \mathbf{V}_1^* = \mathbf{M}, }

which is the desired result:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \mathbf{U} \mathbf \Sigma \mathbf{V}^*. }

Notice the argument could begin with diagonalizing ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*} ⁠ rather than ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠ (This shows directly that ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠ have the same non-zero eigenvalues).

Based on variational characterization

The singular values can also be characterized as the maxima of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u^\mathrm{T} \mathbf M \mathbf v,} ⁠ considered as a function of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V,} ⁠ over particular subspaces. The singular vectors are the values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V} ⁠ where these maxima are attained.

Let ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ denote an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix with real entries. Let ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{k-1} } ⁠ be the unit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (k-1)} -sphere in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^k } , and define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(\mathbf{u}, \mathbf{v}) = \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{v},} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u} \in S^{m-1},} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{v} \in S^{n-1}.}

Consider the function ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} ⁠ restricted to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{m-1} \times S^{n-1}.} ⁠ Since both ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{m-1} } ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{n-1} } ⁠ are compact sets, their product is also compact. Furthermore, since ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma} ⁠ is continuous, it attains a largest value for at least one pair of vectors ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{m-1} } ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v} ⁠ in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S^{n-1}.} ⁠ This largest value is denoted ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1} ⁠ and the corresponding vectors are denoted ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v_1.} ⁠ Since ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1} ⁠ is the largest value of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(\mathbf u, \mathbf v)} ⁠ it must be non-negative. If it were negative, changing the sign of either ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1} ⁠ or ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v_1} ⁠ would make it positive and therefore larger.

Statement. ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1} ⁠ and ⁠ $\mathbf {v} _{1}$ ⁠ are left and right-singular vectors of ⁠ $\mathbf {M}$ ⁠ with corresponding singular value ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1.} ⁠

Proof. Similar to the eigenvalues case, by assumption the two vectors satisfy the Lagrange multiplier equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla \sigma = \nabla \mathbf{u}^\operatorname{T} \mathbf{M} \mathbf{v} - \lambda_1 \cdot \nabla \mathbf{u}^\operatorname{T} \mathbf{u} - \lambda_2 \cdot \nabla \mathbf{v}^\operatorname{T} \mathbf{v} }

After some algebra, this becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{M} \mathbf{v}_1 &= 2 \lambda_1 \mathbf{u}_1 + 0, \\ \mathbf{M}^\operatorname{T} \mathbf{u}_1 &= 0 + 2 \lambda_2 \mathbf{v}_1. \end{align}}

Multiplying the first equation from left by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1^\textrm{T} } ⁠ and the second equation from left by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v_1^\textrm{T} } ⁠ and taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \mathbf u \| = \| \mathbf v \| = 1} into account gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_1 = 2\lambda_1 = 2\lambda_2. }

Plugging this into the pair of equations above, we have

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathbf{M} \mathbf{v}_1 &= \sigma_1 \mathbf{u}_1, \\ \mathbf{M}^\operatorname{T} \mathbf{u}_1 &= \sigma_1 \mathbf{v}_1. \end{align}}

This proves the statement.

More singular vectors and singular values can be found by maximizing ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(\mathbf u, \mathbf v)} ⁠ over normalized ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v} ⁠ which are orthogonal to ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf u_1} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf v_1,} ⁠ respectively.

The passage from real to complex is similar to the eigenvalue case.

Calculating the SVD

One-sided Jacobi algorithm

One-sided Jacobi algorithm is an iterative algorithm,^[21] where a matrix is iteratively transformed into a matrix with orthogonal columns. The elementary iteration is given as a Jacobi rotation,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M\leftarrow MJ(p, q, \theta), }

where the angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} of the Jacobi rotation matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J(p,q,\theta)} is chosen such that after the rotation the columns with numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q} become orthogonal. The indices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p,q)} are swept cyclically, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (p=1\dots m,q=p+1\dots m)} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is the number of columns.

After the algorithm has converged, the singular value decomposition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M=USV^T} is recovered as follows: the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is the accumulation of Jacobi rotation matrices, the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} is given by normalising the columns of the transformed matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} , and the singular values are given as the norms of the columns of the transformed matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .

Two-sided Jacobi algorithm

Two-sided Jacobi SVD algorithm—a generalization of the Jacobi eigenvalue algorithm—is an iterative algorithm where a square matrix is iteratively transformed into a diagonal matrix. If the matrix is not square the QR decomposition is performed first and then the algorithm is applied to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} matrix. The elementary iteration zeroes a pair of off-diagonal elements by first applying a Givens rotation to symmetrize the pair of elements and then applying a Jacobi transformation to zero them,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M \leftarrow J^TGMJ }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is the Givens rotation matrix with the angle chosen such that the given pair of off-diagonal elements become equal after the rotation, and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} is the Jacobi transformation matrix that zeroes these off-diagonal elements. The iterations proceeds exactly as in the Jacobi eigenvalue algorithm: by cyclic sweeps over all off-diagonal elements.

After the algorithm has converged the resulting diagonal matrix contains the singular values. The matrices Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} are accumulated as follows: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U\leftarrow UG^TJ} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V\leftarrow VJ} .

Numerical approach

The singular value decomposition can be computed using the following observations:

The left-singular vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ are a set of orthonormal eigenvectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*} ⁠.
The right-singular vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ are a set of orthonormal eigenvectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠.
The non-zero singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ (found on the diagonal entries of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma} ) are the square roots of the non-zero eigenvalues of both ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*} ⁠.

The SVD of a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is typically computed by a two-step procedure. In the first step, the matrix is reduced to a bidiagonal matrix. This takes order ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(mn^2)} ⁠ floating-point operations (flop), assuming that ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \geq n.} ⁠ The second step is to compute the SVD of the bidiagonal matrix. This step can only be done with an iterative method (as with eigenvalue algorithms). However, in practice it suffices to compute the SVD up to a certain precision, like the machine epsilon. If this precision is considered constant, then the second step takes ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(n)} ⁠ iterations, each costing ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(n)} ⁠ flops. Thus, the first step is more expensive, and the overall cost is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle O(mn^2)} ⁠ flops (Trefethen & Bau III 1997, Lecture 31).

The first step can be done using Householder reflections for a cost of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4mn^2 - 4n^3/3} ⁠ flops, assuming that only the singular values are needed and not the singular vectors. If ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} ⁠ is much larger than ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ⁠ then it is advantageous to first reduce the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ to a triangular matrix with the QR decomposition and then use Householder reflections to further reduce the matrix to bidiagonal form; the combined cost is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2mn^2 + 2n^3} ⁠ flops (Trefethen & Bau III 1997, Lecture 31).

The second step can be done by a variant of the QR algorithm for the computation of eigenvalues, which was first described by Golub & Kahan (1965). The LAPACK subroutine DBDSQR^[22] implements this iterative method, with some modifications to cover the case where the singular values are very small (Demmel & Kahan 1990). Together with a first step using Householder reflections and, if appropriate, QR decomposition, this forms the DGESVD^[23] routine for the computation of the singular value decomposition.

The same algorithm is implemented in the GNU Scientific Library (GSL). The GSL also offers an alternative method that uses a one-sided Jacobi orthogonalization in step 2 (GSL Team 2007). This method computes the SVD of the bidiagonal matrix by solving a sequence of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2} ⁠ SVD problems, similar to how the Jacobi eigenvalue algorithm solves a sequence of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2} ⁠ eigenvalue methods (Golub & Van Loan 1996, §8.6.3). Yet another method for step 2 uses the idea of divide-and-conquer eigenvalue algorithms (Trefethen & Bau III 1997, Lecture 31).

There is an alternative way that does not explicitly use the eigenvalue decomposition.^[24] Usually the singular value problem of a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is converted into an equivalent symmetric eigenvalue problem such as ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \mathbf M^*,} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M,} ⁠ or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} \mathbf{0} & \mathbf{M} \\ \mathbf{M}^* & \mathbf{0} \end{bmatrix}. }

The approaches that use eigenvalue decompositions are based on the QR algorithm, which is well-developed to be stable and fast. Note that the singular values are real and right- and left- singular vectors are not required to form similarity transformations. One can iteratively alternate between the QR decomposition and the LQ decomposition to find the real diagonal Hermitian matrices. The QR decomposition gives ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \Rightarrow \mathbf Q \mathbf R} ⁠ and the LQ decomposition of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R} ⁠ gives ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf R \Rightarrow \mathbf L \mathbf P^*.} ⁠ Thus, at every iteration, we have ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \Rightarrow \mathbf Q \mathbf L \mathbf P^*,} ⁠ update ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M \Leftarrow \mathbf L} ⁠ and repeat the orthogonalizations. Eventually,^{[clarification needed]} this iteration between QR decomposition and LQ decomposition produces left- and right- unitary singular matrices. This approach cannot readily be accelerated, as the QR algorithm can with spectral shifts or deflation. This is because the shift method is not easily defined without using similarity transformations. However, this iterative approach is very simple to implement, so is a good choice when speed does not matter. This method also provides insight into how purely orthogonal/unitary transformations can obtain the SVD.

Analytic result of 2 × 2 SVD

The singular values of a ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \times 2} ⁠ matrix can be found analytically. Let the matrix be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = z_0\mathbf{I} + z_1\sigma_1 + z_2\sigma_2 + z_3\sigma_3}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_i \in \mathbb{C}} are complex numbers that parameterize the matrix, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf I} ⁠ is the identity matrix, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} denote the Pauli matrices. Then its two singular values are given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \sigma_\pm &= \sqrt{|z_0|^2 + |z_1|^2 + |z_2|^2 + |z_3|^2 \pm \sqrt{\bigl(|z_0|^2 + |z_1|^2 + |z_2|^2 + |z_3|^2\bigr)^2 - |z_0^2 - z_1^2 - z_2^2 - z_3^2|^2}} \\ &= \sqrt{|z_0|^2 + |z_1|^2 + |z_2|^2 + |z_3|^2 \pm 2\sqrt{(\operatorname{Re}z_0z_1^*)^2 + (\operatorname{Re}z_0z_2^*)^2 + (\operatorname{Re}z_0z_3^*)^2 + (\operatorname{Im}z_1z_2^*)^2 + (\operatorname{Im}z_2z_3^*)^2 + (\operatorname{Im}z_3z_1^*)^2}} \end{align}}

Reduced SVDs

In applications it is quite unusual for the full SVD, including a full unitary decomposition of the null-space of the matrix, to be required. Instead, it is often sufficient (as well as faster, and more economical for storage) to compute a reduced version of the SVD. The following can be distinguished for an ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times n} ⁠ matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ of rank ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠:

Thin SVD

The thin, or economy-sized, SVD of a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is given by^[25]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \mathbf{U}_k \mathbf \Sigma_k \mathbf{V}^*_k, }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = \min(m, n),} the matrices ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_k} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_k} ⁠ contain only the first ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ⁠ columns of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V,} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma_k} ⁠ contains only the first ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ⁠ singular values from ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma.} ⁠ The matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_k} ⁠ is thus ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times k,} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma_k} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \times k} ⁠ diagonal, and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_k^*} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \times n.} ⁠

The thin SVD uses significantly less space and computation time if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \ll \max(m, n).} ⁠ The first stage in its calculation will usually be a QR decomposition of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ which can make for a significantly quicker calculation in this case.

Compact SVD

The compact SVD of a matrix ⁠ $\mathbf {M}$ ⁠ is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \mathbf U_r \mathbf \Sigma_r \mathbf V_r^*. }

Only the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠ column vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠ row vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ corresponding to the non-zero singular values ⁠ $\mathbf {\Sigma } _{r}$ ⁠ are calculated. The remaining vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ are not calculated. This is quicker and more economical than the thin SVD if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \ll \min(m,n).} ⁠ The matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_r} ⁠ is thus ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times r,} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma_r} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \times r} ⁠ diagonal, and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_r^*} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \times n.} ⁠

Truncated SVD

In many applications the number ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ⁠ of the non-zero singular values is large making even the Compact SVD impractical to compute. In such cases, the smallest singular values may need to be truncated to compute only ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \ll r} ⁠ non-zero singular values. The truncated SVD is no longer an exact decomposition of the original matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ but rather provides the optimal low-rank matrix approximation ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\mathbf M} } ⁠ by any matrix of a fixed rank ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} ⁠

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde{\mathbf{M}} = \mathbf{U}_t \mathbf \Sigma_t \mathbf{V}_t^*, }

where matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U_t} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \times t,} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma_t} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \times t} ⁠ diagonal, and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V_t^*} ⁠ is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \times n.} ⁠ Only the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} ⁠ column vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} ⁠ row vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V^*} ⁠ corresponding to the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} ⁠ largest singular values ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf \Sigma_t} ⁠ are calculated. This can be much quicker and more economical than the compact SVD if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \ll r,} ⁠ but requires a completely different toolset of numerical solvers.

In applications that require an approximation to the Moore–Penrose inverse of the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ the smallest singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ are of interest, which are more challenging to compute compared to the largest ones.

Truncated SVD is employed in latent semantic indexing.^[26]

Norms

Ky Fan norms

The sum of the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ⁠ largest singular values of ⁠ $\mathbf {M}$ ⁠ is a matrix norm, the Ky Fan ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ⁠-norm of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠^[27]

The first of the Ky Fan norms, the Ky Fan 1-norm, is the same as the operator norm of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ as a linear operator with respect to the Euclidean norms of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^m} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K^n.} ⁠ In other words, the Ky Fan 1-norm is the operator norm induced by the standard Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell^2} Euclidean inner product. For this reason, it is also called the operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ on (possibly infinite-dimensional) Hilbert spaces

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \mathbf M \| = \| \mathbf M^* \mathbf M \|^\frac{1}{2} }

But, in the matrix case, ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbf M^* \mathbf M)^{1/2} } ⁠ is a normal matrix, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\mathbf M^* \mathbf M\|^{1/2} } is the largest eigenvalue of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbf M^* \mathbf M)^{1/2},} ⁠ i.e. the largest singular value of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠

The last of the Ky Fan norms, the sum of all singular values, is the trace norm (also known as the 'nuclear norm'), defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \mathbf M \| = \operatorname{Tr}(\mathbf M^* \mathbf M)^{1/2}} (the eigenvalues of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠ are the squares of the singular values).

Hilbert–Schmidt norm

The singular values are related to another norm on the space of operators. Consider the Hilbert–Schmidt inner product on the ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} ⁠ matrices, defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \mathbf{M}, \mathbf{N} \rangle = \operatorname{tr} \left( \mathbf{N}^*\mathbf{M} \right). }

So the induced norm is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \mathbf{M} \| = \sqrt{\langle \mathbf{M}, \mathbf{M} \rangle} = \sqrt{\operatorname{tr} \left( \mathbf{M}^*\mathbf{M} \right)}. }

Since the trace is invariant under unitary equivalence, this shows

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \| \mathbf{M} \| = \sqrt{\vphantom\bigg|\sum_i \sigma_i ^2} }

where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ are the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ This is called the Frobenius norm, Schatten 2-norm, or Hilbert–Schmidt norm of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ Direct calculation shows that the Frobenius norm of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = (m_{ij})} ⁠ coincides with:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\vphantom\bigg|\sum_{ij} | m_{ij} |^2}. }

In addition, the Frobenius norm and the trace norm (the nuclear norm) are special cases of the Schatten norm.

Variations and generalizations

Scale-invariant SVD

The singular values of a matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A} ⁠ are uniquely defined and are invariant with respect to left and/or right unitary transformations of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ In other words, the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U \mathbf A \mathbf V,} ⁠ for unitary matrices ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V,} ⁠ are equal to the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ This is an important property for applications in which it is necessary to preserve Euclidean distances and invariance with respect to rotations.

The Scale-Invariant SVD, or SI-SVD,^[28] is analogous to the conventional SVD except that its uniquely-determined singular values are invariant with respect to diagonal transformations of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ In other words, the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D \mathbf A \mathbf E,} ⁠ for invertible diagonal matrices ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf D} ⁠ and ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf E,} ⁠ are equal to the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ This is an important property for applications for which invariance to the choice of units on variables (e.g., metric versus imperial units) is needed.

Bounded operators on Hilbert spaces

The factorization ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U \mathbf \Sigma \mathbf V^*} ⁠ can be extended to a bounded operator ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ on a separable Hilbert space ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H.} ⁠ Namely, for any bounded operator ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M,} ⁠ there exist a partial isometry ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U,} ⁠ a unitary ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V,} ⁠ a measure space ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X, \mu),} ⁠ and a non-negative measurable ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} ⁠ such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} = \mathbf{U} T_f \mathbf{V}^* }

where ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} ⁠ is the multiplication by ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} ⁠ on ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2(X, \mu).} ⁠

This can be shown by mimicking the linear algebraic argument for the matrix case above. ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V T_f \mathbf V^*} ⁠ is the unique positive square root of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M,} ⁠ as given by the Borel functional calculus for self-adjoint operators. The reason why ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U} ⁠ need not be unitary is that, unlike the finite-dimensional case, given an isometry ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_1} ⁠ with nontrivial kernel, a suitable ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_2} ⁠ may not be found such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} U_1 \\ U_2 \end{bmatrix} }

is a unitary operator.

As for matrices, the singular value factorization is equivalent to the polar decomposition for operators: we can simply write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M = \mathbf U \mathbf V^* \cdot \mathbf V T_f \mathbf V^* }

and notice that ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf U \mathbf V^*} ⁠ is still a partial isometry while ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf V T_f \mathbf V^*} ⁠ is positive.

Singular values and compact operators

The notion of singular values and left/right-singular vectors can be extended to compact operator on Hilbert space as they have a discrete spectrum. If ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} ⁠ is compact, every non-zero ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda} ⁠ in its spectrum is an eigenvalue. Furthermore, a compact self-adjoint operator can be diagonalized by its eigenvectors. If ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is compact, so is ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠. Applying the diagonalization result, the unitary image of its positive square root ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_f} ⁠ has a set of orthonormal eigenvectors ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{e_i\} } ⁠ corresponding to strictly positive eigenvalues ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\sigma_i\} } ⁠. For any ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi} ⁠ in ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H,} ⁠

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{M} \psi = \mathbf{U} T_f \mathbf{V}^* \psi = \sum_i \left \langle \mathbf{U} T_f \mathbf{V}^* \psi, \mathbf{U} e_i \right \rangle \mathbf{U} e_i = \sum_i \sigma_i \left \langle \psi, \mathbf{V} e_i \right \rangle \mathbf{U} e_i, }

where the series converges in the norm topology on ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H.} ⁠ Notice how this resembles the expression from the finite-dimensional case. ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i} ⁠ are called the singular values of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠ ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf U e_i\} } ⁠ (resp. ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\mathbf U e_i\} } ⁠) can be considered the left-singular (resp. right-singular) vectors of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M.} ⁠

Compact operators on a Hilbert space are the closure of finite-rank operators in the uniform operator topology. The above series expression gives an explicit such representation. An immediate consequence of this is:

Theorem. ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M} ⁠ is compact if and only if ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf M^* \mathbf M} ⁠ is compact.

History

The singular value decomposition was originally developed by differential geometers, who wished to determine whether a real bilinear form could be made equal to another by independent orthogonal transformations of the two spaces it acts on. Eugenio Beltrami and Camille Jordan discovered independently, in 1873 and 1874 respectively, that the singular values of the bilinear forms, represented as a matrix, form a complete set of invariants for bilinear forms under orthogonal substitutions. James Joseph Sylvester also arrived at the singular value decomposition for real square matrices in 1889, apparently independently of both Beltrami and Jordan. Sylvester called the singular values the canonical multipliers of the matrix ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf A.} ⁠ The fourth mathematician to discover the singular value decomposition independently is Autonne in 1915, who arrived at it via the polar decomposition. The first proof of the singular value decomposition for rectangular and complex matrices seems to be by Carl Eckart and Gale J. Young in 1936;^[29] they saw it as a generalization of the principal axis transformation for Hermitian matrices.

In 1907, Erhard Schmidt defined an analog of singular values for integral operators (which are compact, under some weak technical assumptions); it seems he was unaware of the parallel work on singular values of finite matrices. This theory was further developed by Émile Picard in 1910, who is the first to call the numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_k} singular values (or in French, valeurs singulières).

Practical methods for computing the SVD date back to Kogbetliantz in 1954–1955 and Hestenes in 1958,^[30] resembling closely the Jacobi eigenvalue algorithm, which uses plane rotations or Givens rotations. However, these were replaced by the method of Gene Golub and William Kahan published in 1965,^[31] which uses Householder transformations or reflections. In 1970, Golub and Christian Reinsch^[32] published a variant of the Golub/Kahan algorithm that is still the one most-used today.

Notes

^ DeAngelis, G. C.; Ohzawa, I.; Freeman, R. D. (October 1995). "Receptive-field dynamics in the central visual pathways". Trends Neurosci. 18 (10): 451–8. doi:10.1016/0166-2236(95)94496-R. PMID 8545912. S2CID 12827601.
^ Depireux, D. A.; Simon, J. Z.; Klein, D. J.; Shamma, S. A. (March 2001). "Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex". J. Neurophysiol. 85 (3): 1220–34. doi:10.1152/jn.2001.85.3.1220. PMID 11247991.
^ The Singular Value Decomposition in Symmetric (Lowdin) Orthogonalization and Data Compression
^ Sahidullah, Md.; Kinnunen, Tomi (March 2016). "Local spectral variability features for speaker verification". Digital Signal Processing. 50: 1–11. doi:10.1016/j.dsp.2015.10.011.
^ Mademlis, Ioannis; Tefas, Anastasios; Pitas, Ioannis (2018). "Regularized SVD-Based Video Frame Saliency for Unsupervised Activity Video Summarization". 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE. pp. 2691–2695. doi:10.1109/ICASSP.2018.8462274. ISBN 978-1-5386-4658-8. S2CID 52286352. Retrieved 19 January 2023.
^ O. Alter, P. O. Brown and D. Botstein (September 2000). "Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling". PNAS. 97 (18): 10101–10106. Bibcode:2000PNAS...9710101A. doi:10.1073/pnas.97.18.10101. PMC 27718. PMID 10963673.
^ O. Alter; G. H. Golub (November 2004). "Integrative Analysis of Genome-Scale Data by Using Pseudoinverse Projection Predicts Novel Correlation Between DNA Replication and RNA Transcription". PNAS. 101 (47): 16577–16582. Bibcode:2004PNAS..10116577A. doi:10.1073/pnas.0406767101. PMC 534520. PMID 15545604.
^ O. Alter; G. H. Golub (August 2006). "Singular Value Decomposition of Genome-Scale mRNA Lengths Distribution Reveals Asymmetry in RNA Gel Electrophoresis Band Broadening". PNAS. 103 (32): 11828–11833. Bibcode:2006PNAS..10311828A. doi:10.1073/pnas.0604756103. PMC 1524674. PMID 16877539.
^ Bertagnolli, N. M.; Drake, J. A.; Tennessen, J. M.; Alter, O. (November 2013). "SVD Identifies Transcript Length Distribution Functions from DNA Microarray Data and Reveals Evolutionary Forces Globally Affecting GBM Metabolism". PLOS ONE. 8 (11): e78913. Bibcode:2013PLoSO...878913B. doi:10.1371/journal.pone.0078913. PMC 3839928. PMID 24282503. Highlight.
^ Edelman, Alan (1992). "On the distribution of a scaled condition number" (PDF). Math. Comp. 58 (197): 185–190. Bibcode:1992MaCom..58..185E. doi:10.1090/S0025-5718-1992-1106966-2.
^ Shen, Jianhong (Jackie) (2001). "On the singular values of Gaussian random matrices". Linear Alg. Appl. 326 (1–3): 1–14. doi:10.1016/S0024-3795(00)00322-0.
^ Walton, S.; Hassan, O.; Morgan, K. (2013). "Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions". Applied Mathematical Modelling. 37 (20–21): 8930–8945. doi:10.1016/j.apm.2013.04.025.
^ Setyawati, Y.; Ohme, F.; Khan, S. (2019). "Enhancing gravitational waveform model through dynamic calibration". Physical Review D. 99 (2): 024010. arXiv:1810.07060. Bibcode:2019PhRvD..99b4010S. doi:10.1103/PhysRevD.99.024010. S2CID 118935941.
^ Sarwar, Badrul; Karypis, George; Konstan, Joseph A. & Riedl, John T. (2000). "Application of Dimensionality Reduction in Recommender System – A Case Study" (PDF). University of Minnesota. {{cite journal}}: Cite journal requires |journal= (help)
^ Bosagh Zadeh, Reza; Carlsson, Gunnar (2013). "Dimension Independent Matrix Square Using MapReduce" (PDF). arXiv:1304.1467. Bibcode:2013arXiv1304.1467B. {{cite journal}}: Cite journal requires |journal= (help)
^ Hadi Fanaee Tork; João Gama (September 2014). "Eigenspace method for spatiotemporal hotspot detection". Expert Systems. 32 (3): 454–464. arXiv:1406.3506. Bibcode:2014arXiv1406.3506F. doi:10.1111/exsy.12088. S2CID 15476557.
^ Hadi Fanaee Tork; João Gama (May 2015). "EigenEvent: An Algorithm for Event Detection from Complex Data Streams in Syndromic Surveillance". Intelligent Data Analysis. 19 (3): 597–616. arXiv:1406.3496. doi:10.3233/IDA-150734. S2CID 17966555.
^ Muralidharan, Vivek; Howell, Kathleen (2023). "Stretching directions in cislunar space: Applications for departures and transfer design". Astrodynamics. 7 (2): 153–178. Bibcode:2023AsDyn...7..153M. doi:10.1007/s42064-022-0147-z. S2CID 252637213.
^ Muralidharan, Vivek; Howell, Kathleen (2022). "Leveraging stretching directions for stationkeeping in Earth-Moon halo orbits". Advances in Space Research. 69 (1): 620–646. Bibcode:2022AdSpR..69..620M. doi:10.1016/j.asr.2021.10.028. S2CID 239490016.
^ To see this, we just have to notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Tr}(\mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2) = \|\mathbf{M} \mathbf{V}_2\|^2} , and remember that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|A\| = 0 \Leftrightarrow A = 0} .
^ Rijk, P.P.M. de (1989). "A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer". SIAM J. Sci. Stat. Comput. 10: 359.
^ Netlib.org
^ Netlib.org
^ mathworks.co.kr/matlabcentral/fileexchange/12674-simple-svd
^ Demmel, James (2000). "Decompositions". Templates for the Solution of Algebraic Eigenvalue Problems. By Bai, Zhaojun; Demmel, James; Dongarra, Jack J.; Ruhe, Axel; van der Vorst, Henk A. Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898719581. ISBN 978-0-89871-471-5.
^ Chicco, D; Masseroli, M (2015). "Software suite for gene and protein annotation prediction and similarity search". IEEE/ACM Transactions on Computational Biology and Bioinformatics. 12 (4): 837–843. doi:10.1109/TCBB.2014.2382127. hdl:11311/959408. PMID 26357324. S2CID 14714823.
^ Fan, Ky. (1951). "Maximum properties and inequalities for the eigenvalues of completely continuous operators". Proceedings of the National Academy of Sciences of the United States of America. 37 (11): 760–766. Bibcode:1951PNAS...37..760F. doi:10.1073/pnas.37.11.760. PMC 1063464. PMID 16578416.
^ Uhlmann, Jeffrey (2018), A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations (PDF), SIAM Journal on Matrix Analysis, vol. 239, pp. 781–800, archived from the original (PDF) on 17 June 2019
^ Eckart, C.; Young, G. (1936). "The approximation of one matrix by another of lower rank". Psychometrika. 1 (3): 211–8. doi:10.1007/BF02288367. S2CID 10163399.
^ Hestenes, M. R. (1958). "Inversion of Matrices by Biorthogonalization and Related Results". Journal of the Society for Industrial and Applied Mathematics. 6 (1): 51–90. doi:10.1137/0106005. JSTOR 2098862. MR 0092215.
^ (Golub & Kahan 1965)
^ Golub, G. H.; Reinsch, C. (1970). "Singular value decomposition and least squares solutions". Numerische Mathematik. 14 (5): 403–420. doi:10.1007/BF02163027. MR 1553974. S2CID 123532178.

References

Banerjee, Sudipto; Roy, Anindya (2014), Linear Algebra and Matrix Analysis for Statistics, Texts in Statistical Science (1st ed.), Chapman and Hall/CRC, ISBN 978-1420095388
Bisgard, James (2021). Analysis and Linear Algebra: The Singular Value Decomposition and Applications. Student Mathematical Library (1st ed.). AMS. ISBN 978-1-4704-6332-8.
Chicco, D; Masseroli, M (2015). "Software suite for gene and protein annotation prediction and similarity search". IEEE/ACM Transactions on Computational Biology and Bioinformatics. 12 (4): 837–843. doi:10.1109/TCBB.2014.2382127. hdl:11311/959408. PMID 26357324. S2CID 14714823.
Trefethen, Lloyd N.; Bau III, David (1997). Numerical linear algebra. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 978-0-89871-361-9.
Demmel, James; Kahan, William (1990). "Accurate singular values of bidiagonal matrices". SIAM Journal on Scientific and Statistical Computing. 11 (5): 873–912. CiteSeerX 10.1.1.48.3740. doi:10.1137/0911052.
Golub, Gene H.; Kahan, William (1965). "Calculating the singular values and pseudo-inverse of a matrix". Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis. 2 (2): 205–224. Bibcode:1965SJNA....2..205G. doi:10.1137/0702016. JSTOR 2949777.
Golub, Gene H.; Van Loan, Charles F. (1996). Matrix Computations (3rd ed.). Johns Hopkins. ISBN 978-0-8018-5414-9.
GSL Team (2007). "§14.4 Singular Value Decomposition". GNU Scientific Library. Reference Manual.
Halldor, Bjornsson and Venegas, Silvia A. (1997). "A manual for EOF and SVD analyses of climate data". McGill University, CCGCR Report No. 97-1, Montréal, Québec, 52pp.
Hansen, P. C. (1987). "The truncated SVD as a method for regularization". BIT. 27 (4): 534–553. doi:10.1007/BF01937276. S2CID 37591557.
Horn, Roger A.; Johnson, Charles R. (1985). "Section 7.3". Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6.
Horn, Roger A.; Johnson, Charles R. (1991). "Chapter 3". Topics in Matrix Analysis. Cambridge University Press. ISBN 978-0-521-46713-1.
Samet, H. (2006). Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann. ISBN 978-0-12-369446-1.
Strang G. (1998). "Section 6.7". Introduction to Linear Algebra (3rd ed.). Wellesley-Cambridge Press. ISBN 978-0-9614088-5-5.
Stewart, G. W. (1993). "On the Early History of the Singular Value Decomposition". SIAM Review. 35 (4): 551–566. CiteSeerX 10.1.1.23.1831. doi:10.1137/1035134. hdl:1903/566. JSTOR 2132388.
Wall, Michael E.; Rechtsteiner, Andreas; Rocha, Luis M. (2003). "Singular value decomposition and principal component analysis". In D.P. Berrar; W. Dubitzky; M. Granzow (eds.). A Practical Approach to Microarray Data Analysis. Norwell, MA: Kluwer. pp. 91–109.
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.6", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

External links

Online SVD calculator

[1] DeAngelis, G. C.; Ohzawa, I.; Freeman, R. D. (October 1995). "Receptive-field dynamics in the central visual pathways". Trends Neurosci. 18 (10): 451–8. doi:10.1016/0166-2236(95)94496-R. PMID 8545912. S2CID 12827601.

[2] Depireux, D. A.; Simon, J. Z.; Klein, D. J.; Shamma, S. A. (March 2001). "Spectro-temporal response field characterization with dynamic ripples in ferret primary auditory cortex". J. Neurophysiol. 85 (3): 1220–34. doi:10.1152/jn.2001.85.3.1220. PMID 11247991.

[3] The Singular Value Decomposition in Symmetric (Lowdin) Orthogonalization and Data Compression

[4] Sahidullah, Md.; Kinnunen, Tomi (March 2016). "Local spectral variability features for speaker verification". Digital Signal Processing. 50: 1–11. doi:10.1016/j.dsp.2015.10.011.

[Mademlis2018-5] Mademlis, Ioannis; Tefas, Anastasios; Pitas, Ioannis (2018). "Regularized SVD-Based Video Frame Saliency for Unsupervised Activity Video Summarization". 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE. pp. 2691–2695. doi:10.1109/ICASSP.2018.8462274. ISBN 978-1-5386-4658-8. S2CID 52286352. Retrieved 19 January 2023.

[6] O. Alter, P. O. Brown and D. Botstein (September 2000). "Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling". PNAS. 97 (18): 10101–10106. Bibcode:2000PNAS...9710101A. doi:10.1073/pnas.97.18.10101. PMC 27718. PMID 10963673.

[7] O. Alter; G. H. Golub (November 2004). "Integrative Analysis of Genome-Scale Data by Using Pseudoinverse Projection Predicts Novel Correlation Between DNA Replication and RNA Transcription". PNAS. 101 (47): 16577–16582. Bibcode:2004PNAS..10116577A. doi:10.1073/pnas.0406767101. PMC 534520. PMID 15545604.

[8] O. Alter; G. H. Golub (August 2006). "Singular Value Decomposition of Genome-Scale mRNA Lengths Distribution Reveals Asymmetry in RNA Gel Electrophoresis Band Broadening". PNAS. 103 (32): 11828–11833. Bibcode:2006PNAS..10311828A. doi:10.1073/pnas.0604756103. PMC 1524674. PMID 16877539.

[9] Bertagnolli, N. M.; Drake, J. A.; Tennessen, J. M.; Alter, O. (November 2013). "SVD Identifies Transcript Length Distribution Functions from DNA Microarray Data and Reveals Evolutionary Forces Globally Affecting GBM Metabolism". PLOS ONE. 8 (11): e78913. Bibcode:2013PLoSO...878913B. doi:10.1371/journal.pone.0078913. PMC 3839928. PMID 24282503. Highlight.

[10] Edelman, Alan (1992). "On the distribution of a scaled condition number" (PDF). Math. Comp. 58 (197): 185–190. Bibcode:1992MaCom..58..185E. doi:10.1090/S0025-5718-1992-1106966-2.

[11] Shen, Jianhong (Jackie) (2001). "On the singular values of Gaussian random matrices". Linear Alg. Appl. 326 (1–3): 1–14. doi:10.1016/S0024-3795(00)00322-0.

[12] Walton, S.; Hassan, O.; Morgan, K. (2013). "Reduced order modelling for unsteady fluid flow using proper orthogonal decomposition and radial basis functions". Applied Mathematical Modelling. 37 (20–21): 8930–8945. doi:10.1016/j.apm.2013.04.025.

[13] Setyawati, Y.; Ohme, F.; Khan, S. (2019). "Enhancing gravitational waveform model through dynamic calibration". Physical Review D. 99 (2): 024010. arXiv:1810.07060. Bibcode:2019PhRvD..99b4010S. doi:10.1103/PhysRevD.99.024010. S2CID 118935941.

[14] Sarwar, Badrul; Karypis, George; Konstan, Joseph A. & Riedl, John T. (2000). "Application of Dimensionality Reduction in Recommender System – A Case Study" (PDF). University of Minnesota. {{cite journal}}: Cite journal requires |journal= (help)

[15] Bosagh Zadeh, Reza; Carlsson, Gunnar (2013). "Dimension Independent Matrix Square Using MapReduce" (PDF). arXiv:1304.1467. Bibcode:2013arXiv1304.1467B. {{cite journal}}: Cite journal requires |journal= (help)

[16] Hadi Fanaee Tork; João Gama (September 2014). "Eigenspace method for spatiotemporal hotspot detection". Expert Systems. 32 (3): 454–464. arXiv:1406.3506. Bibcode:2014arXiv1406.3506F. doi:10.1111/exsy.12088. S2CID 15476557.

[17] Hadi Fanaee Tork; João Gama (May 2015). "EigenEvent: An Algorithm for Event Detection from Complex Data Streams in Syndromic Surveillance". Intelligent Data Analysis. 19 (3): 597–616. arXiv:1406.3496. doi:10.3233/IDA-150734. S2CID 17966555.

[muralidharan2023stretching-18] Muralidharan, Vivek; Howell, Kathleen (2023). "Stretching directions in cislunar space: Applications for departures and transfer design". Astrodynamics. 7 (2): 153–178. Bibcode:2023AsDyn...7..153M. doi:10.1007/s42064-022-0147-z. S2CID 252637213.

[Muralidharan2021-19] Muralidharan, Vivek; Howell, Kathleen (2022). "Leveraging stretching directions for stationkeeping in Earth-Moon halo orbits". Advances in Space Research. 69 (1): 620–646. Bibcode:2022AdSpR..69..620M. doi:10.1016/j.asr.2021.10.028. S2CID 239490016.

[20] To see this, we just have to notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Tr}(\mathbf{V}_2^* \mathbf{M}^* \mathbf{M} \mathbf{V}_2) = \|\mathbf{M} \mathbf{V}_2\|^2} , and remember that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|A\| = 0 \Leftrightarrow A = 0} .

[21] Rijk, P.P.M. de (1989). "A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer". SIAM J. Sci. Stat. Comput. 10: 359.

[22] Netlib.org

[23] Netlib.org

[24] thworks.co.kr/matlabcentral/fileexchange/12674-simple-svd

[25] Demmel, James (2000). "Decompositions". Templates for the Solution of Algebraic Eigenvalue Problems. By Bai, Zhaojun; Demmel, James; Dongarra, Jack J.; Ruhe, Axel; van der Vorst, Henk A. Society for Industrial and Applied Mathematics. doi:10.1137/1.9780898719581. ISBN 978-0-89871-471-5.

[26] Chicco, D; Masseroli, M (2015). "Software suite for gene and protein annotation prediction and similarity search". IEEE/ACM Transactions on Computational Biology and Bioinformatics. 12 (4): 837–843. doi:10.1109/TCBB.2014.2382127. hdl:11311/959408. PMID 26357324. S2CID 14714823.

[27] Fan, Ky. (1951). "Maximum properties and inequalities for the eigenvalues of completely continuous operators". Proceedings of the National Academy of Sciences of the United States of America. 37 (11): 760–766. Bibcode:1951PNAS...37..760F. doi:10.1073/pnas.37.11.760. PMC 1063464. PMID 16578416.

[28] Uhlmann, Jeffrey (2018), A Generalized Matrix Inverse that is Consistent with Respect to Diagonal Transformations (PDF), SIAM Journal on Matrix Analysis, vol. 239, pp. 781–800, archived from the original (PDF) on 17 June 2019

[29] Eckart, C.; Young, G. (1936). "The approximation of one matrix by another of lower rank". Psychometrika. 1 (3): 211–8. doi:10.1007/BF02288367. S2CID 10163399.

[30] Hestenes, M. R. (1958). "Inversion of Matrices by Biorthogonalization and Related Results". Journal of the Society for Industrial and Applied Mathematics. 6 (1): 51–90. doi:10.1137/0106005. JSTOR 2098862. MR 0092215.

[31] (Golub & Kahan 1965)

[32] Golub, G. H.; Reinsch, C. (1970). "Singular value decomposition and least squares solutions". Numerische Mathematik. 14 (5): 403–420. doi:10.1007/BF02163027. MR 1553974. S2CID 123532178.

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v t e Numerical linear algebra
Key concepts	Floating point Numerical stability
Problems	System of linear equations Matrix decompositions Matrix multiplication (algorithms) Matrix splitting Sparse problems
Hardware	CPU cache TLB Cache-oblivious algorithm SIMD Multiprocessing
Software	ATLAS MATLAB Basic Linear Algebra Subprograms (BLAS) LAPACK Specialized libraries General purpose software

v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem