tensor product

English

Noun

tensor product (plural tensor products)

1. (mathematics) The most general bilinear operation in various contexts (as with vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, modules, and so on), denoted by ⊗.
   • 2004, David S. Dummit with Richard M. Foote, chapter 11, in Abstract Algebra‎^[1], →ISBN, →OCLC, 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 $v_{1},...,v_{n}$ and $w_{1},...,w_{m}$ respectively. Then $V\otimes _{F}W$ is a vector space over F of dimension nm with basis $v_{i}\otimes w_{j}$, $1\leq i\leq n$ and $1\leq j\leq m$.

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

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

Usage notes

• The ⊗ symbol can be read out as “tensor”.

Hyponyms

• Kronecker product
• exterior product, wedge product

Meronyms

• tensor (if the tensor product is between algebraic structures)

Holonyms

• monoidal category

Translations

bilinear operation

• Estonian: tensorkorrutis
• Finnish: tensoritulo
• French: produit tensoriel (fr) m
• German: Tensorprodukt n
• Japanese: テンソル積 (tensoru-seki)
• Romanian: produs tensorial n