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, with the following property: for each, if all of the variables but are held constant, then is a linear function of.
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 has a characteristic different from two, else the former two coincide.

Examples

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 , 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, and.
A basis for each is Let
where. In other words, the constant is a function value at one of the eight possible triples of basis vectors, 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 matrix over a commutative ring with identity, as a function of the rows of the matrix. Let be such a matrix and, be the rows of. Then the multilinear function can be written as
satisfying
If we let represent the th row of the identity matrix, we can express each row as the sum
Using the multilinearity of we rewrite as
Continuing this substitution for each we get, for,
where, since in our case,
is a series of nested summations.
Therefore, is uniquely determined by how 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