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 .

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 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 Ck function, then the th derivative of  at each point  in its domain can be viewed as a symmetric -linear function .

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 , 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 (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  matrix over a commutative ring  with identity, as a function of the rows (or equivalently the columns) 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 ,

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

See also
 Algebraic form
 Multilinear form
 Homogeneous polynomial
 Homogeneous function
 Tensors

References

Multilinear algebra