Multilinear map

For multilinear maps used in cryptography, see Cryptographic multilinear map.

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

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

where ${\displaystyle V_{1},\ldots ,V_{n}}$ (${\displaystyle n\in \mathbb {Z} _{\geq 0}}$) and ${\displaystyle W}$ are vector spaces (or modules over a commutative ring), with the following property: for each ${\displaystyle i}$, if all of the variables but ${\displaystyle v_{i}}$ are held constant, then ${\displaystyle f(v_{1},\ldots ,v_{i},\ldots ,v_{n})}$ is a linear function of ${\displaystyle v_{i}}$. ^[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 ${\displaystyle 2^{2}}$.

A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, for any nonnegative integer ${\displaystyle k}$, 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

• Any bilinear map is a multilinear map. For example, any inner product on a ${\displaystyle \mathbb {R} }$-vector space is a multilinear map, as is the cross product of vectors in ${\displaystyle \mathbb {R} ^{3}}$.
• The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
• If ${\displaystyle F\colon \mathbb {R} ^{m}\to \mathbb {R} ^{n}}$ is a C^k function, then the ${\displaystyle k}$th derivative of ${\displaystyle F}$ at each point ${\displaystyle p}$ in its domain can be viewed as a symmetric ${\displaystyle k}$-linear function ${\displaystyle D^{k}\!F\colon \mathbb {R} ^{m}\times \cdots \times \mathbb {R} ^{m}\to \mathbb {R} ^{n}}$.^{[ citation needed ]}

Coordinate representation

Let

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

be a multilinear map between finite-dimensional vector spaces, where ${\displaystyle V_{i}\!}$ has dimension ${\displaystyle d_{i}\!}$, and ${\displaystyle W\!}$ has dimension ${\displaystyle d\!}$. If we choose a basis ${\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}}$ for each ${\displaystyle V_{i}\!}$ and a basis ${\displaystyle \{{\textbf {b}}_{1},\ldots ,{\textbf {b}}_{d}\}}$ for ${\displaystyle W\!}$ (using bold for vectors), then we can define a collection of scalars ${\displaystyle A_{j_{1}\cdots j_{n}}^{k}}$ by

${\displaystyle f({\textbf {e}}_{1j_{1}},\ldots ,{\textbf {e}}_{nj_{n}})=A_{j_{1}\cdots j_{n}}^{1}\,{\textbf {b}}_{1}+\cdots +A_{j_{1}\cdots j_{n}}^{d}\,{\textbf {b}}_{d}.}$

Then the scalars ${\displaystyle \{A_{j_{1}\cdots j_{n}}^{k}\mid 1\leq j_{i}\leq d_{i},1\leq k\leq d\}}$ completely determine the multilinear function ${\displaystyle f\!}$. In particular, if

${\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{d_{i}}v_{ij}{\textbf {e}}_{ij}\!}$

for ${\displaystyle 1\leq i\leq n\!}$, then

${\displaystyle f({\textbf {v}}_{1},\ldots ,{\textbf {v}}_{n})=\sum _{j_{1}=1}^{d_{1}}\cdots \sum _{j_{n}=1}^{d_{n}}\sum _{k=1}^{d}A_{j_{1}\cdots j_{n}}^{k}v_{1j_{1}}\cdots v_{nj_{n}}{\textbf {b}}_{k}.}$

Example

Let's take a trilinear function

${\displaystyle g\colon R^{2}\times R^{2}\times R^{2}\to R,}$

where V_i = R², d_i = 2, i = 1,2,3, and W = R, d = 1.

A basis for each V_i is ${\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.}$ Let

${\displaystyle g({\textbf {e}}_{1i},{\textbf {e}}_{2j},{\textbf {e}}_{3k})=f({\textbf {e}}_{i},{\textbf {e}}_{j},{\textbf {e}}_{k})=A_{ijk},}$

where ${\displaystyle i,j,k\in \{1,2\}}$. In other words, the constant ${\displaystyle A_{ijk}}$ is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three ${\displaystyle V_{i}}$), namely:

${\displaystyle \{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.}$

Each vector ${\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}}$ can be expressed as a linear combination of the basis vectors

${\displaystyle {\textbf {v}}_{i}=\sum _{j=1}^{2}v_{ij}{\textbf {e}}_{ij}=v_{i1}\times {\textbf {e}}_{1}+v_{i2}\times {\textbf {e}}_{2}=v_{i1}\times (1,0)+v_{i2}\times (0,1).}$

The function value at an arbitrary collection of three vectors ${\displaystyle {\textbf {v}}_{i}\in R^{2}}$ can be expressed as

${\displaystyle g({\textbf {v}}_{1},{\textbf {v}}_{2},{\textbf {v}}_{3})=\sum _{i=1}^{2}\sum _{j=1}^{2}\sum _{k=1}^{2}A_{ijk}v_{1i}v_{2j}v_{3k},}$

or in expanded form as

${\displaystyle {\begin{aligned}g((a,b),(c,d)&,(e,f))=ace\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1})+acf\times g({\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+ade\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1})+adf\times g({\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2})+bce\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1})+bcf\times g({\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2})\\&+bde\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1})+bdf\times g({\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}).\end{aligned}}}$

Relation to tensor products

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

${\displaystyle f\colon V_{1}\times \cdots \times V_{n}\to W{\text{,}}}$

and linear maps

${\displaystyle F\colon V_{1}\otimes \cdots \otimes V_{n}\to W{\text{,}}}$

where ${\displaystyle V_{1}\otimes \cdots \otimes V_{n}\!}$ denotes the tensor product of ${\displaystyle V_{1},\ldots ,V_{n}}$. The relation between the functions ${\displaystyle f}$ and ${\displaystyle F}$ is given by the formula

${\displaystyle f(v_{1},\ldots ,v_{n})=F(v_{1}\otimes \cdots \otimes v_{n}).}$

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 a_i, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as

${\displaystyle D(A)=D(a_{1},\ldots ,a_{n}),}$

satisfying

${\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}$

If we let ${\displaystyle {\hat {e}}_{j}}$ represent the jth row of the identity matrix, we can express each row a_i as the sum

${\displaystyle a_{i}=\sum _{j=1}^{n}A(i,j){\hat {e}}_{j}.}$

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

${\displaystyle D(A)=D\left(\sum _{j=1}^{n}A(1,j){\hat {e}}_{j},a_{2},\ldots ,a_{n}\right)=\sum _{j=1}^{n}A(1,j)D({\hat {e}}_{j},a_{2},\ldots ,a_{n}).}$

Continuing this substitution for each a_i we get, for 1 ≤ i ≤ n,

${\displaystyle D(A)=\sum _{1\leq k_{1}\leq n}\ldots \sum _{1\leq k_{i}\leq n}\ldots \sum _{1\leq k_{n}\leq n}A(1,k_{1})A(2,k_{2})\dots A(n,k_{n})D({\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}).}$

Therefore, D(A) is uniquely determined by how D operates on ${\displaystyle {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}}$.

Example

In the case of 2×2 matrices, we get

${\displaystyle D(A)=A_{1,1}A_{1,2}D({\hat {e}}_{1},{\hat {e}}_{1})+A_{1,1}A_{2,2}D({\hat {e}}_{1},{\hat {e}}_{2})+A_{1,2}A_{2,1}D({\hat {e}}_{2},{\hat {e}}_{1})+A_{1,2}A_{2,2}D({\hat {e}}_{2},{\hat {e}}_{2}),\,}$

where ${\displaystyle {\hat {e}}_{1}=[1,0]}$ and ${\displaystyle {\hat {e}}_{2}=[0,1]}$. If we restrict ${\displaystyle D}$ to be an alternating function, then ${\displaystyle D({\hat {e}}_{1},{\hat {e}}_{1})=D({\hat {e}}_{2},{\hat {e}}_{2})=0}$ and ${\displaystyle D({\hat {e}}_{2},{\hat {e}}_{1})=-D({\hat {e}}_{1},{\hat {e}}_{2})=-D(I)}$. Letting ${\displaystyle D(I)=1}$, we get the determinant function on 2×2 matrices:

${\displaystyle D(A)=A_{1,1}A_{2,2}-A_{1,2}A_{2,1}.}$

Properties

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

See also

• Algebraic form
• Multilinear form
• Homogeneous polynomial
• Homogeneous function
• Tensors

References

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.