tensor 積
tensor product
tensor 積とは雙線形寫像$ \otimes:V\times W\to V\otimes Wであって、任意の雙線形寫像$ h:V\times W\to Zに對して、可換圖式$ V\times W\xrightarrow{\otimes}V\otimes W\xrightarrow{\tilde h}Z\xleftarrow{h}V\times W,$ h=\otimes;\tilde hを成り立たせる線形寫像$ \tilde hがただ一つ存在するものを言ふ 可換$ V\otimes W\cong W\otimes V
結合$ V\otimes(W\otimes Z)\cong(V\otimes W)\otimes Z
線形寫像$ f:V\to X,$ g:W\to Yに對して$ f\otimes g:V\otimes W\to X\otimes Y,v\otimes w\mapsto f(v)\otimes g(w)と定義する code:tex
A\otimes B=
\begin{pmatrix}
a_{11} & \cdots & a_{1n} \\
\vdots & \ddots & \vdots \\
a_{m1} & \cdots & a_{mn}
\end{pmatrix}\otimes
\begin{pmatrix}
b_{11} & \cdots & b_{1q} \\
\vdots & \ddots & \vdots \\
b_{p1} & \cdots & b_{pq}
\end{pmatrix} :=
\begin{pmatrix}
a_{11}b_{11} & \cdots & a_{11}b_{1q} & \cdots & a_{1n}b_{11} & \cdots & a_{1n}b_{1q} \\
\vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{11}b_{p1} & \cdots & a_{11}b_{pq} & \cdots & a_{1n}b_{p1} & \cdots & a_{1n}b_{pq} \\
\vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{m1}b_{11} & \cdots & a_{m1}b_{1q} & \cdots & a_{mn}b_{11} & \cdots & a_{mn}b_{1q} \\
\vdots & \ddots & \vdots & \ddots & \vdots & \ddots & \vdots \\
a_{m1}b_{p1} & \cdots & a_{m1}b_{pq} & \cdots & a_{mn}b_{p1} & \cdots & a_{mn}b_{pq} \\
\end{pmatrix}
tensor 代數
tensor 空閒