Flight dynamics (fixed-wing aircraft)

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

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

where $V_1,\dots ,V_n$ and $W$ are vector spaces (or modules), with the following property: for each $i$, if all of the variables but $v_i$ are held constant, then $f(v_1,\dots ,v_n)$ is a linear function of $v_i$.

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.

If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps. The latter 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 vector space is a multilinear map, as is the cross product of vectors in ${\mathbb{ℝ}}^3$.
• The determinant of a matrix is an antisymmetric multilinear function of the columns (or rows) of a square matrix.
• If $F:{\mathbb{ℝ}}^m\to {\mathbb{ℝ}}^n$ is a C^k function, then the $k$th derivative of $F$ at each point $p$ in its domain can be viewed as a symmetric $k$-linear function $D^{k}f(p):{\mathbb{ℝ}}^m\times \cdots \times {\mathbb{ℝ}}^m\to {\mathbb{ℝ}}^n$.
• The tensor-to-vector projection in multilinear subspace learning is a multilinear map as well.

Coordinate representation

Let

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

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 $\{{\mathbf{\text{e}}}_{i1},\dots ,{\mathbf{\text{e}}}_{i{d}_i}\}$ for each $V_i$ and a basis $\{{\mathbf{\text{b}}}_1,\dots ,{\mathbf{\text{b}}}_d\}$ for $W$ (using bold for vectors), then we can define a collection of scalars $A_{j_1\cdots j_n}^k$ by

$f({\mathbf{\text{e}}}_{1j_1},\dots ,{\mathbf{\text{e}}}_{n{j}_n})=A_{j_1\cdots j_n}^1{\mathbf{\text{b}}}_1+\cdots +A_{j_1\cdots j_n}^d{\mathbf{\text{b}}}_d.$

Then the scalars $\{A_{j_1\cdots j_n}^k\mid 1\le j_i\le d_i,1\le k\le d\}$ completely determine the multilinear function $f$. In particular, if

${\mathbf{\text{v}}}_i=\sum _{j=1}^{d_i}{v}_{ij}{\mathbf{\text{e}}}_{ij}$

for $1\le i\le n$, then

$f({\mathbf{\text{v}}}_1,\dots ,{\mathbf{\text{v}}}_n)=\sum _{j_1=1}^{d_1}\cdots \sum _{j_n=1}^{d_n}\sum _{k=1}^d{A}_{j_1\cdots j_n}^k{v}_{1j_1}\cdots v_{n{j}_n}{\mathbf{\text{b}}}_k.$

Relation to tensor products

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

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

and linear maps

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

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

$F(v_1\otimes \cdots \otimes v_n)=f(v_1,\dots ,v_n).$

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

$D(A)=D(a_1,\dots ,a_n)$

satisfying

$D(a_1,\dots ,c{a}_i+{a_{i^{\prime }}}^{\prime },\dots ,a_n)=cD(a_1,\dots ,a_i,\dots ,a_n)+D(a_1,\dots ,{a_{i^{\prime }}}^{\prime },\dots ,a_n)$

If we let ${\hat{e}}_j$ represent the jth row of the identity matrix we can express each row $a_i$ 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

$D(A)=D(\sum _{j=1}^{n}A(1,j){\hat{e}}_j,a_2,\dots ,a_n)=\sum _{j=1}^{n}A(1,j)D({\hat{e}}_j,a_2,\dots ,a_n)$

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

$D(A)=\sum _{1\le k_i\le n}A(1,k_1)A(2,k_2)\dots A(n,k_n)D({\hat{e}}_{k_1},\dots ,{\hat{e}}_{k_n})$

where, since in our case $1\le i\le n$

$\sum _{1\le k_i\le n}=\sum _{1\le k_1\le n}\dots \sum _{1\le k_i\le n}\dots \sum _{1\le k_n\le n}$

as a series of nested summations.

Therefore, D(A) is uniquely determined by how $D$ operates on ${\hat{e}}_{k_1},\dots ,{\hat{e}}_{k_n}$.

Example

In the case of 2×2 matrices we get

$D(A)=A_{1,1}{A}_{2,1}D({\hat{e}}_1,{\hat{e}}_1)+A_{1,1}{A}_{2,2}D({\hat{e}}_1,{\hat{e}}_2)+A_{1,2}{A}_{2,1}D({\hat{e}}_2,{\hat{e}}_1)+A_{1,2}{A}_{2,2}D({\hat{e}}_2,{\hat{e}}_2)$

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.

For n>1, the only n-linear map which is also a linear map is the zero function, see bilinear map#Examples.

See also

• Algebraic form
• Multilinear form
• Homogeneous polynomial
• Homogeneous function
• Tensors
• Multilinear projection
• Multilinear subspace learning