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 W are vector spaces (or modules over a commutative ring), with the following property: for each i, if all of the variables but v_i are held constant, then is a linear function of v_i. 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 2^2. A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer k, 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.

Examples

Coordinate representation

Let be a multilinear map between finite-dimensional vector spaces, where V_i! has dimension d_i!, and W! has dimension d!. If we choose a basis for each V_i! and a basis for W! (using bold for vectors), then we can define a collection of scalars by Then the scalars completely determine the multilinear function f!. In particular, if for, then

Example

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 A_{i j k} is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three V_i), 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 f and F 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 satisfying If we let \hat{e}_j 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.

Example

In the case of 2×2 matrices, we get where and. If we restrict D to be an alternating function, then and. Letting D(I) = 1, we get the determinant function on 2×2 matrices:

Properties