Multilinear map

• 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 square matrix is a multilinear function of the columns (or rows); it is also an alternating function of the columns (or rows). • 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. ==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 , and . A basis for each 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 matrix over a commutative ring with identity, as a function of the rows (or equivalently the columns) of the matrix. Let be such a matrix and , be the rows of . Then the multilinear function 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 th row of the identity matrix, we can express each row as the sum :a_{i} = \sum_{j=1}^n A(i,j)\hat{e}_{j}. Using the multilinearity of we rewrite 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 we get, for , : 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, is uniquely determined by how 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==