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Multilinear map

{{Short description|Vector-valued function of multiple vectors, linear in each argument}}

{{For|multilinear maps used in cryptography|Cryptographic multilinear map}}

{{More citations needed|date=October 2023}}

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

:f\colon V_1 \times \cdots \times V_n \to W\text{,}

where V_1,\ldots,V_n (n\in\mathbb Z_{\ge0}) and W are vector spaces (or modules over a commutative ring), with the following property: for each i, if all of the variables but v_i are held constant, then f(v_1, \ldots,

v_i, \ldots, v_n) is a linear function of v_i.{{cite book |author-link=Serge Lang |first=Serge |last=Lang |title=Algebra |chapter=XIII. Matrices and Linear Maps §S Determinants |chapter-url=https://books.google.com/books?id=Fge-BwqhqIYC&pg=PA511 |date=2005 |orig-date=2002 |publisher=Springer |edition=3rd |isbn=978-0-387-95385-4 |pages=511– |volume=211 |series=Graduate Texts in Mathematics}} 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 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 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 \mathbb R-vector space is a multilinear map, as is the cross product of vectors in \mathbb{R}^3.
  • The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix.
  • If F\colon \mathbb{R}^m \to \mathbb{R}^n is a Ck function, then the kth derivative of F at each point p in its domain can be viewed as a symmetric k-linear function D^k\!F\colon \mathbb{R}^m\times\cdots\times\mathbb{R}^m \to \mathbb{R}^n.{{Citation needed|date=October 2023}}

Coordinate representation

Let

:f\colon V_1 \times \cdots \times V_n \to W\text{,}

be a multilinear map between finite-dimensional vector spaces, where V_i\! has dimension d_i\!, and W\! has dimension d\!. If we choose a basis \{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} for each V_i\! and a basis \{\textbf{b}_1,\ldots,\textbf{b}_d\} for W\! (using bold for vectors), then we can define a collection of scalars A_{j_1\cdots j_n}^k by

: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 \{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 f\!. In particular, if

:\textbf{v}_i = \sum_{j=1}^{d_i} v_{ij} \textbf{e}_{ij}\!

for 1 \leq i \leq n\!, then

: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

:g\colon R^2 \times R^2 \times R^2 \to R,

where {{math|1=Vi = R2, di = 2, i = 1,2,3}}, and {{math|1=W = R, d = 1}}.

A basis for each {{mvar|Vi}} is \{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} = \{\textbf{e}_{1}, \textbf{e}_{2}\} = \{(1,0), (0,1)\}. Let

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

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

:

\{\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 \textbf{v}_i \in V_i = R^2 can be expressed as a linear combination of the basis vectors

:\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 \textbf{v}_i \in R^2 can be expressed as

:g(\textbf{v}_1,\textbf{v}_2, \textbf{v}_3) = \sum_{i=1}^{2} \sum_{j=1}^{2} \sum_{k=1}^{2} A_{i j k} v_{1i} v_{2j} v_{3k},

or in expanded form as

: \begin{align}

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{align}

Relation to tensor products

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

:f\colon V_1 \times \cdots \times V_n \to W\text{,}

and linear maps

:F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,}

where V_1 \otimes \cdots \otimes V_n\! denotes the tensor product of V_1,\ldots,V_n. The relation between the functions f and F is given by the formula

: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 {{math|n×n}} matrix over a commutative ring {{mvar|K}} with identity, as a function of the rows (or equivalently the columns) of the matrix. Let {{math|A}} be such a matrix and {{math|ai, 1 ≤ i ≤ n}}, be the rows of {{math|A}}. Then the multilinear function {{math|D}} can be written as

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

satisfying

:D(a_{1},\ldots,c a_{i} + a_{i}',\ldots,a_{n}) = c D(a_{1},\ldots,a_{i},\ldots,a_{n}) + D(a_{1},\ldots,a_{i}',\ldots,a_{n}).

If we let \hat{e}_j represent the {{mvar|j}}th row of the identity matrix, we can express each row {{math|ai}} as the sum

:a_{i} = \sum_{j=1}^n A(i,j)\hat{e}_{j}.

Using the multilinearity of {{math|D}} we rewrite {{math|D(A)}} as

:

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 {{math|ai}} we get, for {{math|1 ≤ i ≤ n}},

:

D(A) = \sum_{1\le k_1 \le n} \ldots \sum_{1\le k_i \le n} \ldots \sum_{1\le k_n \le 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, {{math|D(A)}} is uniquely determined by how {{mvar|D}} operates on \hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}.

Example

In the case of 2×2 matrices, we get

:

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 \hat{e}_1 = [1,0] and \hat{e}_2 = [0,1]. If we restrict D to be an alternating function, then D(\hat{e}_1,\hat{e}_1) = D(\hat{e}_2,\hat{e}_2) = 0 and D(\hat{e}_2,\hat{e}_1) = -D(\hat{e}_1,\hat{e}_2) = -D(I). Letting D(I) = 1, we get the determinant function on 2×2 matrices:

: 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

References

Category:Multilinear algebra