tensor productとは 意味・読み方・使い方

名詞

tensor product (複数形 tensor products)

1. (mathematics) The most general bilinear operation in various contexts (as with vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, modules, かつ so on), denoted by ⊗.
   • 2004, David S. Dummit; Richard M. Foote, chapter 11, in Abstract Algebra‎^[1], →ISBN, OCLC 758130105, page 421:

     Linear Transformations on Tensor Products of Vector Spaces

     [...]

     Proposition 16. Let V and W be finite dimensional vector spaces over the field F with bases ${\displaystyle v_{1},...,v_{n}}$ and ${\displaystyle w_{1},...,w_{m}}$ respectively. Then ${\displaystyle V\otimes _{F}W}$ is a vector space over F of dimension nm with basis ${\displaystyle v_{i}\otimes w_{j}}$, ${\displaystyle 1\leq i\leq n}$ and ${\displaystyle 1\leq j\leq m}$.

   • 2012, Lecture 1 . Hopf Algebras and Combinatorics (Federico Ardila)‎^[2], YouTube, Federico Ardila, spoken by Federico Ardila (Federico Ardila), 27:30 from the start:

     [The tensor product] ${\displaystyle U\otimes V}$ is the span of ${\displaystyle \{u\otimes v:u\in U,v\in V\}}$
         modulo the relations
         ${\displaystyle {\begin{cases}\cdot \ u\otimes (v+v')=u\otimes v+u\otimes v'\\\cdot \ (u+u')\otimes v=u\otimes v+u'\otimes v\\\cdot \ (\lambda u)\otimes v=\lambda (u\otimes v)=u\otimes (\lambda v)\end{cases}}}$