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Multilinear map

Vector-valued function of multiple vectors, linear in each argument

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated July 25, 2026

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} _{\geq 0}) 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}. 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.

01Examples

02Coordinate 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}.

03Example

Let's take a trilinear function

g\colon R^{2}\times R^{2}\times R^{2}\to R, where Vi = R2, di = 2, i = 1,2,3, and W = R, d = 1.

A basis for each 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_{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 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_{ijk}v_{1i}v_{2j}v_{3k}, or in expanded form as {\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}}

04Relation 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}).

05Multilinear 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 ai, 1 ≤ i ≤ n, be the rows of A. Then the multilinear function D can be written as

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

satisfying

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 {\hat {e}}_{j} represent the jth row of the identity matrix, we can express each row ai as the sum

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

Using the multilinearity of D we rewrite 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 ai we get, for 1 ≤ i ≤ n,

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 {\hat {e}}_{k_{1}},\dots ,{\hat {e}}_{k_{n}}.

06Example

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}.

07Properties

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

This article is adapted from the Wikipedia article “Multilinear map”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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