Multilinear map - AbsoluteAstronomy.com

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, a multilinear map is a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of several variables that is linear separately in each variable. More precisely, a multilinear map is a function

where

and

are vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s (or modules

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

), with the following property: for each

, if all of the variables but

are held constant, then

is a linear function of

.

A multilinear map 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

Multilinear algebra

In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...

.

If all variables belong to the same space, one can consider symmetric

Symmetric function

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

,
antisymmetric

Antisymmetric

The word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry and Asymmetry. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about...

and alternating

Alternating

Alternating may refer to:In mathematics:*alternating form*alternating group*alternating series*alternating knot*alternating mapIn electronics:*alternating current...

k-linear maps. The latter coincide if the underlying ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

(or field) has a characteristic different from two,
else the former two coincide.

Examples

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
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

of vectors in .
The determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

of a matrix is an antisymmetric
Antisymmetric
The word antisymmetric refers to a change to an opposite quantity when another quantity is symmetrically changed. This concept is related to that of Symmetry and Asymmetry. The difference between these three concepts can be simply illustrated with Latin letters. The character "A" is symmetric about...

multilinear function of the columns (or rows) of a square matrix.
If is a C^k function
Smooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

, then the th derivative of at each point in its domain can be viewed as a symmetric
Symmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

-linear function .
The tensor-to-vector projection in multilinear subspace learning
Multilinear subspace learning
Multilinear subspace learning aims to learn a specific small part of a large space of multidimensional objects having a particular desired property. It is a dimensionality reduction approach for finding a low-dimensional representation with certain preferred characteristics of high-dimensional...

is a multilinear map as well.

Coordinate representation

Let

be a multilinear map between finite-dimensional vector spaces, where

has dimension

, and

has dimension

. If we choose a basis

Basis (linear algebra)

In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

for each

and a basis

for

(using bold for vectors), then we can define a collection of scalars

Then the scalars

completely determine the multilinear function

. In particular, if

for

, then

Relation to tensor products

There is a natural one-to-one correspondence between multilinear maps

and linear maps

where

denotes the tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

. The relation between the functions

and

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

Commutative ring

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....

K with identity, as a function of the rows (or equivalently the columns) of the matrix. Let A be such a matrix and

, 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

as the sum

Using the multilinearity of D we rewrite D(A) as

Continuing this substitution for each

we get, for 1 ≤ i ≤ n

where, since in our case

as a series of nested summations.

Therefore, D(A) is uniquely determined by how

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

we get the determinant function on 2×2 matrices:

Properties