In multilinear algebra, applying a map that is the tensor product of linear maps to a tensor is called a multilinear multiplication.
Abstract definition Let F {\displaystyle F} be a field of characteristic zero, such as R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } . Let V k {\displaystyle V_{k}} be a finite-dimensional vector space over F {\displaystyle F} , and let A ∈ V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d {\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 v k ∈ V k {\displaystyle \mathbf {v} _{k}\in V_{k}} such that A = v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d {\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 A k : V k → W k {\displaystyle A_{k}:V_{k}\to W_{k}} , then the multilinear multiplication of A {\displaystyle {\mathcal {A}}} with ( A 1 , A 2 , … , A d ) {\displaystyle (A_{1},A_{2},\ldots ,A_{d})} is defined as the action on A {\displaystyle {\mathcal {A}}} of the tensor product of these linear maps, namely
A 1 ⊗ A 2 ⊗ ⋯ ⊗ A d : V 1 ⊗ V 2 ⊗ ⋯ ⊗ V d → W 1 ⊗ W 2 ⊗ ⋯ ⊗ W d , v 1 ⊗ v 2 ⊗ ⋯ ⊗ v d ↦ A 1 ( v 1 ) ⊗ A 2 ( v 2 ) ⊗ ⋯ ⊗ A d ( v d ) {\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}}}
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