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] 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.

Examples

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  • 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  .[citation needed]

Coordinate representation

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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

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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

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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

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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  .

Example

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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

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

See also

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References

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  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.