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, 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.
- 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 .
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
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 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as
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,
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 .
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.