Multilinear multiplication

In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.

Let ${\displaystyle F}$ be a field of characteristic zero, such as ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$. Let ${\displaystyle V_{k}}$ be a finite-dimensional vector space over ${\displaystyle F}$, and let ${\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}}$ be an order-d simple tensor, i.e., there exist some vectors ${\displaystyle \mathbf {v} _{k}\in V_{k}}$ such that ${\displaystyle {\mathcal {A}}=\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}}$. If we are given a collection of linear maps ${\displaystyle A_{k}:V_{k}\to W_{k}}$, then the multilinear multiplication of ${\displaystyle {\mathcal {A}}}$ with ${\displaystyle (A_{1},A_{2},\ldots ,A_{d})}$ is defined as the action on ${\displaystyle {\mathcal {A}}}$ of the tensor product of these linear maps, namely

$${\displaystyle {\begin{aligned}A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d}:V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}&\to W_{1}\otimes W_{2}\otimes \cdots \otimes W_{d},\\\mathbf {v} _{1}\otimes \mathbf {v} _{2}\otimes \cdots \otimes \mathbf {v} _{d}&\mapsto A_{1}(\mathbf {v} _{1})\otimes A_{2}(\mathbf {v} _{2})\otimes \cdots \otimes A_{d}(\mathbf {v} _{d})\end{aligned}}}$$

Since the tensor product of linear maps is itself a linear map, and because every tensor admits a tensor rank decomposition, the above expression extends linearly to all tensors. That is, for a general tensor ${\displaystyle {\mathcal {A}}\in V_{1}\otimes V_{2}\otimes \cdots \otimes V_{d}}$, the multilinear multiplication is

$${\displaystyle {\begin{aligned}&{\mathcal {B}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})\\[4pt]={}&(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})\left(\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}\right)\\[5pt]={}&\sum _{i=1}^{r}A_{1}(\mathbf {a} _{i}^{1})\otimes A_{2}(\mathbf {a} _{i}^{2})\otimes \cdots \otimes A_{d}(\mathbf {a} _{i}^{d})\end{aligned}}}$$

where ${\textstyle {\mathcal {A}}=\sum _{i=1}^{r}\mathbf {a} _{i}^{1}\otimes \mathbf {a} _{i}^{2}\otimes \cdots \otimes \mathbf {a} _{i}^{d}}$ with ${\displaystyle \mathbf {a} _{i}^{k}\in V_{k}}$ is one of ${\displaystyle {\mathcal {A}}}$'s tensor rank decompositions. The validity of the above expression is not limited to a tensor rank decomposition; in fact, it is valid for any expression of ${\displaystyle {\mathcal {A}}}$ as a linear combination of pure tensors, which follows from the universal property of the tensor product.

It is standard to use the following shorthand notations in the literature for multilinear multiplications:$${\displaystyle (A_{1},A_{2},\ldots ,A_{d})\cdot {\mathcal {A}}:=(A_{1}\otimes A_{2}\otimes \cdots \otimes A_{d})({\mathcal {A}})}$$and$${\displaystyle A_{k}\cdot _{k}{\mathcal {A}}:=(\operatorname {Id} _{V_{1}},\ldots ,\operatorname {Id} _{V_{k-1}},A_{k},\operatorname {Id} _{V_{k+1}},\ldots ,\operatorname {Id} _{V_{d}})\cdot {\mathcal {A}},}$$where ${\displaystyle \operatorname {Id} _{V_{k}}:V_{k}\to V_{k}}$ is the identity operator.

In computational multilinear algebra it is conventional to work in coordinates. Assume that an inner product is fixed on ${\displaystyle V_{k}}$ and let ${\displaystyle V_{k}^{*}}$ denote the dual vector space of ${\displaystyle V_{k}}$. Let ${\displaystyle \{e_{1}^{k},\ldots ,e_{n_{k}}^{k}\}}$ be a basis for ${\displaystyle V_{k}}$, let ${\displaystyle \{(e_{1}^{k})^{*},\ldots ,(e_{n_{k}}^{k})^{*}\}}$ be the dual basis, and let ${\displaystyle \{f_{1}^{k},\ldots ,f_{m_{k}}^{k}\}}$ be a basis for ${\displaystyle W_{k}}$. The linear map ${\textstyle M_{k}=\sum _{i=1}^{m_{k}}\sum _{j=1}^{n_{k}}m_{i,j}^{(k)}f_{i}^{k}\otimes (e_{j}^{k})^{*}}$ is then represented by the matrix ${\displaystyle {\widehat {M}}_{k}=[m_{i,j}^{(k)}]\in F^{m_{k}\times n_{k}}}$. Likewise, with respect to the standard tensor product basis ${\displaystyle \{e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}\}_{j_{1},j_{2},\ldots ,j_{d}}}$, the abstract tensor$${\displaystyle {\mathcal {A}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}e_{j_{1}}^{1}\otimes e_{j_{2}}^{2}\otimes \cdots \otimes e_{j_{d}}^{d}}$$is represented by the multidimensional array ${\displaystyle {\widehat {\mathcal {A}}}=[a_{j_{1},j_{2},\ldots ,j_{d}}]\in F^{n_{1}\times n_{2}\times \cdots \times n_{d}}}$ . Observe that $${\displaystyle {\widehat {\mathcal {A}}}=\sum _{j_{1}=1}^{n_{1}}\sum _{j_{2}=1}^{n_{2}}\cdots \sum _{j_{d}=1}^{n_{d}}a_{j_{1},j_{2},\ldots ,j_{d}}\mathbf {e} _{j_{1}}^{1}\otimes \mathbf {e} _{j_{2}}^{2}\otimes \cdots \otimes \mathbf {e} _{j_{d}}^{d},}$$