Editing 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 (mathematics)|function]] of several variables that is [[Linear map|linear]] separately in each variable.  More precisely, a multilinear map is a function

:<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

where <math>V_1,\ldots,V_n</math> (<math>n\in\mathbb Z_{\ge0}</math>) and <math>W</math> are [[vector space]]s (or [[module (mathematics)|module]]s over a [[commutative ring]]), with the following property: for each <math>i</math>, if all of the variables but <math>v_i</math> are held constant, then <math>f(v_1, \ldots, 
 v_i, \ldots, v_n)</math> is a [[linear function]] of <math>v_i</math>.<ref>{{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}}</ref> 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 <math>2^2</math>.

A multilinear map of one variable is a [[linear map]], and of two variables is a [[bilinear map]].  More generally, for any nonnegative integer <math>k</math>, 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 function|symmetric]], [[Bilinear_form#Symmetric,_skew-symmetric_and_alternating_forms|antisymmetric]] and [[alternating map|alternating]] ''k''-linear maps. The latter two coincide if the underlying [[ring (mathematics)|ring]] (or [[field (mathematics)|field]]) has a [[Characteristic (algebra)|characteristic]] different from two, else the former two coincide.

==Examples==
* Any [[bilinear map]] is a multilinear map.  For example, any [[inner product]] on a <math>\mathbb R</math>-vector space is a multilinear map, as is the [[cross product]] of vectors in <math>\mathbb{R}^3</math>.
* The [[determinant]] of a [[square matrix]] is a multilinear function of the columns (or rows); it is also an [[Alternating form|alternating]] function of the columns (or rows).
* If <math>F\colon \mathbb{R}^m \to \mathbb{R}^n</math> is a [[smooth function|''C<sup>k</sup>'' function]], then the <math>k</math>th derivative of <math>F</math> at each point <math>p</math> in its domain can be viewed as a [[symmetric function|symmetric]] <math>k</math>-linear function <math>D^k\!F\colon \mathbb{R}^m\times\cdots\times\mathbb{R}^m \to \mathbb{R}^n</math>.{{Citation needed|date=October 2023}}

==Coordinate representation==
Let

:<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

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

:<math>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.</math>

Then the scalars <math>\{A_{j_1\cdots j_n}^k \mid 1\leq j_i\leq d_i, 1 \leq k \leq d\}</math> completely determine the multilinear function <math>f\!</math>.  In particular, if

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

for <math>1 \leq i \leq n\!</math>, then

:<math>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.</math>

==Example==
Let's take a trilinear function

:<math>g\colon R^2 \times R^2 \times R^2 \to R, </math>
where {{math|1=''V<sub>i</sub>'' = ''R''<sup>2</sup>, ''d<sub>i</sub>'' = 2, ''i'' = 1,2,3}}, and {{math|1=''W'' = ''R'', ''d'' = 1}}.

A basis for each {{mvar|V<sub>i</sub>}} is <math>\{\textbf{e}_{i1},\ldots,\textbf{e}_{id_i}\} = \{\textbf{e}_{1}, \textbf{e}_{2}\} = \{(1,0), (0,1)\}.</math> Let

:<math>g(\textbf{e}_{1i},\textbf{e}_{2j},\textbf{e}_{3k}) = f(\textbf{e}_{i},\textbf{e}_{j},\textbf{e}_{k}) = A_{ijk},</math>
where <math>i,j,k \in \{1,2\}</math>. In other words, the constant <math>A_{i j k}</math> is a function value at one of the eight possible triples of basis vectors (since there are two choices for each of the three <math>V_i</math>), namely: 
:<math>
\{\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\}.
</math>

Each vector <math>\textbf{v}_i \in V_i = R^2</math> can be expressed as a linear combination of the basis vectors

:<math>\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).</math>

The function value at an arbitrary collection of three vectors <math>\textbf{v}_i \in R^2</math> can be expressed as
:<math>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},</math>
or in expanded form as
:<math> \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} 
</math>

==Relation to tensor products==
There is a [[natural transformation|natural]] one-to-one correspondence between multilinear maps

:<math>f\colon V_1 \times \cdots \times V_n \to W\text{,}</math>

and linear maps

:<math>F\colon V_1 \otimes \cdots \otimes V_n \to W\text{,}</math>

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

:<math>f(v_1,\ldots,v_n)=F(v_1\otimes \cdots \otimes v_n).</math>

==Multilinear functions on ''n''&times;''n'' matrices==
One can consider multilinear functions, on an {{math|''n''&times;''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|''a<sub>i</sub>'', 1 ≤ ''i'' ≤ ''n''}}, be the rows of {{math|''A''}}.  Then the multilinear function {{math|''D''}} can be written as

:<math>D(A) = D(a_{1},\ldots,a_{n}),</math>

satisfying

:<math>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}).</math>

If we let <math>\hat{e}_j</math> represent the {{mvar|j}}th row of the identity matrix, we can express each row {{math|''a<sub>i</sub>''}} as the sum

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

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

:<math>
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).
</math>

Continuing this substitution for each {{math|''a<sub>i</sub>''}} we get, for {{math|1 ≤ ''i'' ≤ ''n''}},

:<math>
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}}).
</math>

Therefore, {{math|''D''(''A'')}} is uniquely determined by how {{mvar|D}} operates on <math>\hat{e}_{k_{1}},\dots,\hat{e}_{k_{n}}</math>.

==Example==
In the case of 2&times;2 matrices, we get

:<math>
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), \,
</math>

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

:<math> D(A) = A_{1,1}A_{2,2} - A_{1,2}A_{2,1} .</math>

==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]]
* [[Tensor]]s

==References==
<references/>

[[Category:Multilinear algebra]]