张量积

张量积：

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

张量积性质：

(1)右进法则：

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

(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)$A=A_{m imes m}, B=B_{n imes n}$，$tr(A otimes B) = tr(A)tr(B)$

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

张量积的用法：

(1)求广义逆：

(i)

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