Multilinear map

Summary

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.

ExamplesEdit

  • 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  .

Coordinate representationEdit

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

 

ExampleEdit

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 productsEdit

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 matricesEdit

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,

 

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

ExampleEdit

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:

 

PropertiesEdit

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

See alsoEdit

ReferencesEdit

  1. ^ Lang, Serge (2005) [2002]. "XIII. Matrices and Linear Maps §S Determinants". Algebra. Graduate Texts in Mathematics. Vol. 211 (3rd ed.). Springer. pp. 511–. ISBN 978-0-387-95385-4.