Analytic function of a matrix

In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size.

This is used for defining the exponential of a matrix, which is involved in the closed-form solution of systems of linear differential equations.

Extending scalar function to matrix functions

There are several techniques for lifting a real function to a square matrix function such that interesting properties are maintained. All of the following techniques yield the same matrix function, but the domains on which the function is defined may differ.

Power series

If the analytic function f has the Taylor expansion Failed to parse (SVG (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(x) = c_0 + c_1 x + c_2 x^2 + \cdots} then a matrix function $A\mapsto f(A)$ can be defined by substituting x by a square matrix: powers become matrix powers, additions become matrix sums and multiplications by coefficients become scalar multiplications. If the series converges for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |x|<r} , then the corresponding matrix series converges for matrices A such that $\|A\|<r$ for some matrix norm that satisfies Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \|AB\|\leq \|A\|\|B\|} .

Diagonalizable matrices

A square matrix A is diagonalizable, if there is an invertible matrix P such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D=P^{-1}\,A\,P} is a diagonal matrix, that is, D has the shape Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle D={\begin{bmatrix}d_{1}&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &d_{n}\end{bmatrix}}.}

As $A=P\,D\,P^{-1},$ it is natural to set

$$f(A)=P\,{\begin{bmatrix}f(d_{1})&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &f(d_{n})\end{bmatrix}}\,P^{-1}.$$

It can be verified that the matrix f(A) does not depend on a particular choice of P.

For example, suppose one is seeking $\Gamma (A)=(A-1)!$ for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A={\begin{bmatrix}1&3\\2&1\end{bmatrix}}.}

One has

$$A=P{\begin{bmatrix}1-{\sqrt {6}}&0\\0&1+{\sqrt {6}}\end{bmatrix}}P^{-1}~,$$

for

$$P={\begin{bmatrix}1/2&1/2\\-{\frac {1}{\sqrt {6}}}&{\frac {1}{\sqrt {6}}}\end{bmatrix}}~.$$

Application of the formula then simply yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma(A) = \begin{bmatrix} 1/2 & 1/2 \\ -\frac{1}{\sqrt{6} } & \frac{1}{\sqrt{6} } \end{bmatrix} \cdot \begin{bmatrix} \Gamma(1-\sqrt{6}) & 0\\ 0&\Gamma(1+\sqrt{6}) \end{bmatrix} \cdot \begin{bmatrix} 1 & -\sqrt{6}/2 \\ 1 & \sqrt{6}/2 \end{bmatrix} \approx \begin{bmatrix} 2.8114 & 0.4080 \\ 0.2720 & 2.8114 \end{bmatrix} ~. }

Likewise, Failed to parse (SVG (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^4 = \begin{bmatrix} 1/2 & 1/2 \\ -\frac{1}{\sqrt{6} } & \frac{1}{\sqrt{6} } \end{bmatrix} \cdot \begin{bmatrix} (1-\sqrt{6})^4 & 0\\ 0&(1+\sqrt{6})^4 \end{bmatrix} \cdot \begin{bmatrix} 1 & -\sqrt{6}/2 \\ 1 & \sqrt{6}/2 \end{bmatrix} = \begin{bmatrix} 73 & 84\\ 56 & 73 \end{bmatrix} ~. }

Jordan decomposition

Main article: Jordan normal form

All complex matrices, whether they are diagonalizable or not, have a Jordan normal 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 A = P\,J\,P^{-1}} , where the matrix J consists of Jordan blocks. Consider these blocks separately and apply the power series to a Jordan block: Failed to parse (SVG (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{bmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \vdots & \vdots \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \cdots & \ddots & \lambda & 1 \\ 0 & \cdots & \cdots & 0 & \lambda \end{bmatrix} \right) = \begin{bmatrix} \frac{f(\lambda)}{0!} & \frac{f'(\lambda)}{1!} & \frac{f''(\lambda)}{2!} & \cdots & \frac{f^{(n-1)}(\lambda)}{(n-1)!} \\ 0 & \frac{f(\lambda)}{0!} & \frac{f'(\lambda)}{1!} & \vdots & \frac{f^{(n-2)}(\lambda)}{(n-2)!} \\ 0 & 0 & \ddots & \ddots & \vdots \\ \vdots & \cdots & \ddots & \frac{f(\lambda)}{0!} & \frac{f'(\lambda)}{1!} \\ 0 & \cdots & \cdots & 0 & \frac{f(\lambda)}{0!} \end{bmatrix}. }

This definition can be used to extend the domain of the matrix function beyond the set of matrices with spectral radius smaller than the radius of convergence of the power series. Note that there is also a connection to divided differences.

A related notion is the Jordan–Chevalley decomposition which expresses a matrix as a sum of a diagonalizable and a nilpotent part.

Hermitian matrices

A Hermitian matrix has all real eigenvalues and can always be diagonalized by a unitary matrix P, according to the spectral theorem. In this case, the Jordan definition is natural. Moreover, this definition allows one to extend standard inequalities for real functions:

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 f(a) \leq g(a)} for all eigenvalues of $A$, 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 f(A) \preceq g(A)} . (As a convention, Failed to parse (SVG (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 \preceq Y \Leftrightarrow Y - X } is a positive-semidefinite matrix.) The proof follows directly from the definition.

Cauchy integral

Cauchy's integral formula from complex analysis can also be used to generalize scalar functions to matrix functions. Cauchy's integral formula states that for any analytic function f defined on a set D ⊂ C, one has Failed to parse (SVG (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(x) = \frac{1}{2\pi i} \oint_{C}\! {\frac{f(z)}{z-x}}\, \mathrm{d}z ~,} where C is a closed simple curve inside the domain D enclosing x.

Now, replace x by a matrix A and consider a path C inside D that encloses all eigenvalues of A. One possibility to achieve this is to let C be a circle around the origin with radius larger than ‖A‖ for an arbitrary matrix norm ‖·‖. Then, f (A) is definable 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(A) = \frac{1}{2\pi i} \oint_C f(z)\left(z I - A\right)^{-1} \mathrm{d}z \,. }

This integral can readily be evaluated numerically using the trapezium rule, which converges exponentially in this case. That means that the precision of the result doubles when the number of nodes is doubled. In routine cases, this is bypassed by Sylvester's formula.

This idea applied to bounded linear operators on a Banach space, which can be seen as infinite matrices, leads to the holomorphic functional calculus.

See also: Sylvester's formula

Matrix perturbations

The above Taylor power series allows the scalar Failed to parse (SVG (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} to be replaced by the matrix. This is not true in general when expanding 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 A(\eta) = A+\eta B} about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \eta = 0} unless Failed to parse (SVG (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,B]=0} . A counterexample 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(x) = x^{3}} , which has a finite length Taylor series. We compute this in two ways,

• Distributive law: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(A+\eta B)=(A+\eta B)^{3}=A^{3}+\eta (A^{2}B+ABA+BA^{2})+\eta ^{2}(AB^{2}+BAB+B^{2}A)+\eta ^{3}B^{3}}
• Using scalar Taylor expansion 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 f(a+\eta b)} and replacing scalars with matrices at the end: Failed to parse (SVG (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} f(a+\eta b) &= f(a) + f'(a)\frac{\eta b}{1!} + f''(a)\frac{(\eta b)^2}{2!} + f'''(a)\frac{(\eta b)^3}{3!} \\[.5em] &= a^3 + 3a^2(\eta b) + 3a(\eta b)^2 + (\eta b)^3 \\[.5em] &\to A^3 = + 3A^2(\eta B) + 3A(\eta B)^2 + (\eta B)^3 \end{align}}

The scalar expression assumes commutativity while the matrix expression does not, and thus they cannot be equated directly unless Failed to parse (SVG (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,B]=0} . For some f(x) this can be dealt with using the same method as scalar Taylor series. For example, ${\textstyle f(x)={\frac {1}{x}}}$. 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 A^{-1}} exists 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 f(A+\eta B) = f(\mathbb{I} + \eta A^{-1}B)f(A)} . The expansion of the first term then follows the power series given 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 f(\mathbb{I} + \eta A^{-1}B) = \mathbb{I} - \eta A^{-1}B + (-\eta A^{-1}B)^2 + \cdots = \sum_{n=0}^\infty (-\eta A^{-1}B)^n }

The convergence criteria of the power series then apply, requiring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Vert \eta A^{-1}B \Vert} to be sufficiently small under the appropriate matrix norm. For more general problems, which cannot be rewritten in such a way that the two matrices commute, the ordering of matrix products produced by repeated application of the Leibniz rule must be tracked.

Arbitrary function of a 2×2 matrix

An arbitrary function f(A) of a 2×2 matrix A has its Sylvester's formula simplify 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 f(A) = \frac{f(\lambda_+) + f(\lambda_-)}{2} I + \frac{A - \left (\frac{tr(A)}{2}\right )I}{\sqrt{\left (\frac{tr(A)}{2}\right)^2 - |A|}} \frac{f(\lambda_+) - f(\lambda_-)}{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 \lambda_\pm} are the eigenvalues of its characteristic equation, |A − λI| = 0, and 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 \lambda_\pm = \frac{tr(A)}{2} \pm \sqrt{\left (\frac{tr(A)}{2}\right )^2 - |A|} .} However, if there is degeneracy, the following formula is used, where f' is the derivative of f.

$$f(A)=f\left({\frac {tr(A)}{2}}\right)I+\mathrm {adj} \left({\frac {tr(A)}{2}}I-A\right)f'\left({\frac {tr(A)}{2}}\right).$$

Examples

• Matrix polynomial
• Matrix root
• Matrix logarithm
• Matrix exponential
• Matrix sign function

Classes of matrix functions

Using the semidefinite ordering (Failed to parse (SVG (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 \preceq Y \Leftrightarrow Y - X} is positive-semidefinite 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 X \prec Y \Leftrightarrow Y - X } is positive definite), some of the classes of scalar functions can be extended to matrix functions of Hermitian matrices.

Operator monotone

Main article: Operator monotone function

A function f is called operator monotone 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 0 \prec A \preceq H \Rightarrow f(A) \preceq f(H) } for all self-adjoint matrices A,H with spectra in the domain of f. This is analogous to monotone function in the scalar case.

Operator concave/convex

A function f is called operator concave 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 \tau f(A) + (1-\tau) f(H) \preceq f \left ( \tau A + (1-\tau)H \right ) } for all self-adjoint matrices A,H with spectra in the domain of f 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 \tau \in [0,1]} . This definition is analogous to a concave scalar function. An operator convex function can be defined be switching Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \preceq} 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 \succeq} in the definition above.

Examples

The matrix log is both operator monotone and operator concave. The matrix square is operator convex. The matrix exponential is none of these. Loewner's theorem states that a function on an open interval is operator monotone if and only if it has an analytic extension to the upper and lower complex half planes so that the upper half plane is mapped to itself.

See also

• Algebraic Riccati equation
• Sylvester's formula
• Loewner order
• Matrix calculus
• Trace inequalities
• Trigonometric functions of matrices

References

• Higham, Nicholas J. (2008). Functions of matrices theory and computation. Philadelphia: Society for Industrial and Applied Mathematics. ISBN 9780898717778.