# Multilinear map

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]

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, 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 coincide if the underlying ring (or field) has a characteristic different from two, else the former two coincide.

## Examples

- Any bilinear map is a multilinear map. For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in .
- The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
- If is a
*C*function, then the th derivative of at each point in its domain can be viewed as a symmetric -linear function .^{k} - The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.

## 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 *V _{i}* =

*R*

^{2},

*d*= 2,

_{i}*i*= 1,2,3, and

*W*=

*R*,

*d*= 1.

A basis for each V_{i} 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

*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 *a _{i}*, 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 *a _{i}* as the sum

Using the multilinearity of *D* we rewrite *D*(*A*) as

Continuing this substitution for each *a _{i}* we get, for 1 ≤

*i*≤

*n*,

where, since in our case 1 ≤ *i* ≤ *n*,

is a series of nested summations.

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.

## See also

## References

- Serge Lang. Algebra. Springer; 3rd edition (January 8, 2002)