张量积

标签：amp   法则   array   time   nbsp   用法   begin   ...   公式   

张量积：

\[A \otimes B = {\left[ {\begin{array}{*{20}{c}}
{{a_{11}}B}&{...}&{{a_{1n}}B}\\
{...}&{...}&{...}\\
{{a_{m1}}B}&{...}&{{a_{mm}}B}
\end{array}} \right]_{m \times p,n \times q}}\]

张量积性质：

(1)右进法则：

\[\left[ {\begin{array}{*{20}{c}}
A&B\\
C&D
\end{array}} \right] \otimes E = \left[ {\begin{array}{*{20}{c}}
{A \otimes E}&{B \otimes E}\\
{C \otimes E}&{D \otimes E}
\end{array}} \right]\]

(2)左进法则不成立

(3)吸收公式：$({A_1} \otimes {B_1})({A_2} \otimes {B_2}) = ({A_1}{A_2} \otimes {B_1}{B_2})$

(4)${(A \otimes B)^H} = {A^H} \otimes {B^H}$

(5)${(A \otimes B)^ + } = {A^ + } \otimes {B^ + }$

(6)${(A \otimes B)^ {-1} } = {A^ {-1} } \otimes {B^ {-1} }$

(7)$A=A_{m \times m}, B=B_{n \times n}$，$tr(A \otimes B) = tr(A)tr(B)$

(8)$A=A_{m \times m}, B=B_{n \times n}$，$\det (A \otimes B) = \det {(A)^n}\det {(B)^m}$

张量积的用法：

(1)求广义逆：

(i)

\[{\left[ {\begin{array}{*{20}{c}}
A&0
\end{array}} \right]^ + } = {(\left[ {\begin{array}{*{20}{c}}
1&0
\end{array}} \right] \otimes A)^ + } = {\left[ {\begin{array}{*{20}{c}}
1&0
\end{array}} \right]^ + } \otimes {A^ + } = \left[ {\begin{array}{*{20}{c}}
1\\
0
\end{array}} \right] \otimes {A^ + } = \left[ {\begin{array}{*{20}{c}}
{{A^ + }}\\
0
\end{array}} \right]\]

张量积

标签：amp   法则   array   time   nbsp   用法   begin   ...   公式   

原文地址：https://www.cnblogs.com/codeDog123/p/10238625.html