Vector-valued function of multiple vectors, linear in each argument
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Revision as of 20:18, 15 January 2024 by 67.249.248.128(talk)(Very minor practical clarification. ✞)
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function
where () and are vector spaces (or modules over a commutative ring), with the following property: for each , if all of the variables but are held constant, then is a linear function of .[1] One way to visualize this is to imagine two orthogonal vectors; if one of these vectors is scaled by a factor of 2 while the other remains unchanged, the cross product likewise scales by a factor of two. If both are scaled by a factor of 2, the cross product scales by a factor of .
A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer , a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra.
If all variables belong to the same space, one can consider symmetric, antisymmetric and alternatingk-linear maps. The latter two coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.
be a multilinear map between finite-dimensional vector spaces, where has dimension , and has dimension . If we choose a basis for each and a basis for (using bold for vectors), then we can define a collection of scalars by
Then the scalars completely determine the multilinear function . In particular, if
for , then
Example
Let's take a trilinear function
where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.
A basis for each Vi is Let
where . In other words, the constant is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three ), namely:
Each vector can be expressed as a linear combination of the basis vectors
The function value at an arbitrary collection of three vectors can be expressed as
or in expanded form as
Relation to tensor products
There is a natural one-to-one correspondence between multilinear maps
and linear maps
where denotes the tensor product of . The relation between the functions and is given by the formula
Multilinear functions on n×n matrices
One can consider multilinear functions, on an n×n matrix over a commutative ringK with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as
satisfying
If we let represent the jth row of the identity matrix, we can express each row ai as the sum
Using the multilinearity of D we rewrite D(A) as
Continuing this substitution for each ai we get, for 1 ≤ i ≤ n,
Therefore, D(A) is uniquely determined by how D operates on .
Example
In the case of 2×2 matrices, we get
where and . If we restrict to be an alternating function, then and . Letting , we get the determinant function on 2×2 matrices:
Properties
A multilinear map has a value of zero whenever one of its arguments is zero.