Multilinear map

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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 .[1]

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.

Examples[edit]

  • 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 antisymmetric 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 .
  • The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.

Coordinate representation[edit]

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[edit]

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[edit]

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[edit]

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 ≤ in, 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 ≤ in,

where, since in our case 1 ≤ in,

is a series of nested summations.

Therefore, D(A) is uniquely determined by how D operates on .

Example[edit]

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[edit]

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

See also[edit]

References[edit]

  1. ^ Lang. Algebra. Springer; 3rd edition (January 8, 2002)