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

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

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

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

Let

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

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

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

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

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

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

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

Let's take a trilinear function

${\displaystyle g\colon R^{2}\times R^{2}\times R^{2}\to R,}${\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 ${\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.}${\displaystyle \{{\textbf {e}}_{i1},\ldots ,{\textbf {e}}_{id_{i}}\}=\{{\textbf {e}}_{1},{\textbf {e}}_{2}\}=\{(1,0),(0,1)\}.} Let

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

where ${\displaystyle i,j,k\in \{1,2\}}${\displaystyle i,j,k\in \{1,2\}}. In other words, the constant ${\displaystyle A_{ijk}}${\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 ${\displaystyle V_{i}}${\displaystyle V_{i}}), namely:

${\displaystyle \{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{1},{\textbf {e}}_{2},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{1},{\textbf {e}}_{2}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{1}\},\{{\textbf {e}}_{2},{\textbf {e}}_{2},{\textbf {e}}_{2}\}.}${\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 ${\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}}${\displaystyle {\textbf {v}}_{i}\in V_{i}=R^{2}} can be expressed as a linear combination of the basis vectors

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

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

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

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

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

and linear maps

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

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

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

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

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

satisfying

${\displaystyle D(a_{1},\ldots ,ca_{i}+a_{i}',\ldots ,a_{n})=cD(a_{1},\ldots ,a_{i},\ldots ,a_{n})+D(a_{1},\ldots ,a_{i}',\ldots ,a_{n}).}${\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 ${\displaystyle {\hat {e}}_{j}}${\displaystyle {\hat {e}}_{j}} represent the jth row of the identity matrix, we can express each row a_i as the sum

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

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

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

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

In the case of 2×2 matrices, we get

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

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

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

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

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

• Multilinear algebra

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