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
are vector spaces (or modules over a commutative ring), with the following property: for each
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.
Let be a multilinear map between finite-dimensional vector spaces, where
(using bold for vectors), then we can define a collection of scalars
completely determine the multilinear function
, then Let's take a trilinear function where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.
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
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 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
is given by the formula 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
to be an alternating function, then
, we get the determinant function on 2×2 matrices: