# 张量积本文重定向自 张量积

${\displaystyle \mathbf {b} \otimes \mathbf {a} \rightarrow {\begin{bmatrix}b_{1}\\b_{2}\\b_{3}\\b_{4}\end{bmatrix}}{\begin{bmatrix}a_{1}&a_{2}&a_{3}\end{bmatrix}}={\begin{bmatrix}a_{1}b_{1}&a_{2}b_{1}&a_{3}b_{1}\\a_{1}b_{2}&a_{2}b_{2}&a_{3}b_{2}\\a_{1}b_{3}&a_{2}b_{3}&a_{3}b_{3}\\a_{1}b_{4}&a_{2}b_{4}&a_{3}b_{4}\end{bmatrix}}}$

## 两个张量的张量积

${\displaystyle (V\otimes U)_{i_{1}i_{2}\dots i_{m+n}}=V_{i_{1}i_{2}i_{3}\dots i_{n}}U_{i_{n+1}i_{n+2}\dots i_{n+m}}}$

${\displaystyle \mathrm {rank} (U\otimes V)=\mathrm {rank} (U)\mathrm {rank} (V)}$

### 例子

U 是类型 (1,1) 的张量，带有分量 Uαβ；并设 V 是类型 (1,0) 的张量，带有分量 Vγ。则

${\displaystyle U^{\alpha }{}_{\beta }V^{\gamma }=(U\otimes V)^{\alpha }{}_{\beta }{}^{\gamma }}$

${\displaystyle V^{\mu }U^{\nu }{}_{\sigma }=(V\otimes U)^{\mu \nu }{}_{\sigma }}$

## 两个矩阵的克罗内克积

${\displaystyle U\otimes V={\begin{bmatrix}u_{11}V&u_{12}V&\cdots \\u_{21}V&u_{22}V\\\vdots &&\ddots \end{bmatrix}}={\begin{bmatrix}u_{11}v_{11}&u_{11}v_{12}&\cdots &u_{12}v_{11}&u_{12}v_{12}&\cdots \\u_{11}v_{21}&u_{11}v_{22}&&u_{12}v_{21}&u_{12}v_{22}\\\vdots &&\ddots \\u_{21}v_{11}&u_{21}v_{12}\\u_{21}v_{21}&u_{21}v_{22}\\\vdots \end{bmatrix}}}$

## 多重线性映射的张量积

${\displaystyle (f\otimes g)(x_{1},\dots ,x_{k+m})=f(x_{1},\dots ,x_{k})g(x_{k+1},\dots ,x_{k+m})}$

## 向量空间的张量积

• ${\displaystyle (v_{1}+v_{2})\otimes w=v_{1}\otimes w+v_{2}\otimes w}$
• ${\displaystyle v\otimes (w_{1}+w_{2})=v\otimes w_{1}+v\otimes w_{2}}$
• ${\displaystyle cv\otimes w=v\otimes cw=c(v\otimes w)}$

${\displaystyle 0v\otimes w=v\otimes 0w=0(v\otimes w)=0}$

## 张量积的泛性质

${\displaystyle \phi (u,w)=u\otimes w}$

${\displaystyle T:V\otimes W\rightarrow X}$

${\displaystyle \psi =T\circ \phi }$

${\displaystyle B(V\times W,X)}$

${\displaystyle L(V\otimes W,X)}$

## 希尔伯特空间的张量积

### 定义

${\displaystyle H_{1}}$${\displaystyle H_{2}}$ 是两个希尔伯特空间，分别带有内积 ${\displaystyle \langle \cdot ,\cdot \rangle _{1}}$${\displaystyle \langle \cdot ,\cdot \rangle _{2}}$。构造 H1H2 的张量积${\displaystyle H_{1}{\hat {\otimes }}H_{2}}$如下：

${\displaystyle \langle \phi _{1}\otimes \phi _{2},\psi _{1}\otimes \psi _{2}\rangle =\langle \phi _{1},\psi _{1}\rangle _{1}\cdot \langle \phi _{2},\psi _{2}\rangle _{2}}$

## 注解

1. ^ 类似的公式对反变以及混合型张量也成立。尽管许多情形，比如定义了一个内积，这种区分是无关的。