Vector-valued function of multiple vectors, linear in each argument
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.
- 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 Ck function, then the
th derivative of
at each point
in its domain can be viewed as a symmetric
-linear function
.[citation needed]
Coordinate representation
[edit]
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

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
[edit]
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
[edit]
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 ≤ 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
.
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:

- A multilinear map has a value of zero whenever one of its arguments is zero.