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), 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 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.
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:
, i = 1,2,3, and .
Basis of all is equal: . Then denote:
- , where . In other words, the constant means a function value at one of 8 possible combinations of basis vectors, one per each :
Each vector can be expressed as a linear combination of the basis vectors:
The function value at an arbitrary collection of 3 vectors can be expressed:
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 , 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 as the sum
Using the multilinearity of D we rewrite D(A) as
Continuing this substitution for each we get, for 1 ≤ i ≤ n
- where, since in our case
- as a series of nested summations.
Therefore, D(A) is uniquely determined by how operates on .
In the case of 2×2 matrices we get
Where and . If we restrict D 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.
For n>1, the only n-linear map which is also a linear map is the zero function, see bilinear map#Examples.
- ↑ Lang. Algebra. Springer; 3rd edition (January 8, 2002)