This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: "Multilinear map" – news · newspapers · books · scholar · JSTOR (October 2023) (Learn how and when to remove this message)

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 alternating k-linear maps. The latter two coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.

Examples

Coordinate representation

Let

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 ring K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and ai, 1 ≤ in, 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 ≤ in,

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

See also

References

  1. ^ Lang, Serge (2005) [2002]. "XIII. Matrices and Linear Maps §S Determinants". Algebra. Graduate Texts in Mathematics. Vol. 211 (3rd ed.). Springer. pp. 511–. ISBN 978-0-387-95385-4.