定义
m × n \mathit{m} \times \mathit{n} m×n矩阵 A = [ a 1 , ⋯ , a n ] \boldsymbol{A}=\left[\boldsymbol{a}_{1}, \cdots, \boldsymbol{a}_{\mathit{n} }\right] A=[a1,⋯,an]和 p × q \mathit{p} \times \mathit{q} p×q矩阵 B \boldsymbol{B} B的 K r o n e c k e r \mathit{Kronecker} Kronecker积记作 A ⊗ B \boldsymbol{A} \otimes \boldsymbol{B} A⊗B，是一个 m p × n q \mathit{mp} \times \mathit{nq} mp×nq矩阵，定义为 A ⊗ B = [ a 1 B , ⋯ , a n B ] = [ a i j B ] i = 1 , j = 1 m , n = [ a 11 B a 12 B ⋯ a 1 n B a 21 B a 22 B ⋯ a 2 n B ⋮ ⋮ ⋱ ⋮ a m 1 B a m 2 B ⋯ a m n B ] \boldsymbol{A} \otimes \boldsymbol{B}=\left[\boldsymbol{a}_{1} \boldsymbol{B}, \cdots, \boldsymbol{a}_{\mathit{n} } \boldsymbol{B}\right]=\left[\mathit{a}_{\mathit{i j} } \boldsymbol{B}\right]_{\mathit{i} =1, \mathit{j} =1}^{\mathit{m} , \mathit{n} }=\left[\begin{array}{cccc} \mathit{a} _{11} \boldsymbol{B} & \mathit{a} _{12} \boldsymbol{B} & \cdots & \mathit{a} _{1 \mathit{n} } \boldsymbol{B} \\ \mathit{a} _{21} \boldsymbol{B} & \mathit{a} _{22} \boldsymbol{B} & \cdots & \mathit{a} _{2 \mathit{n} } \boldsymbol{B} \\ \vdots & \vdots & \ddots & \vdots \\ \mathit{a} _{\mathit{m} 1} \boldsymbol{B} & \mathit{a} _{\mathit{m} 2} \boldsymbol{B} & \cdots & \mathit{a} _{\mathit{m n} } \boldsymbol{B} \end{array}\right] A⊗B=[a1B,⋯,anB]=[aijB]i=1,j=1m,n=⎣⎢⎢⎢⎡a11Ba21B⋮am1Ba12Ba22B⋮am2B⋯⋯⋱⋯a1nBa2nB⋮amnB⎦⎥⎥⎥⎤ K r o n e c k e r \mathit{Kronecker} Kronecker积也称直积或者张量积。文章来源地址https://www.toymoban.com/news/detail-729498.html
K r o n e c k e r \mathit{Kronecker} Kronecker积的等式性质
- 对于矩阵 A m × n \boldsymbol A_{\mathit{m} \times \mathit{n} } Am×n和 B p × q \boldsymbol B_{\mathit{p} \times \mathit{q} } Bp×q，一般有 A ⊗ B ≠ B ⊗ A \boldsymbol A \otimes\boldsymbol B \neq\boldsymbol B \otimes\boldsymbol A A⊗B=B⊗A。
- 任意矩阵与零矩阵的 K r o n e c k e r \mathit{Kronecker} Kronecker积等于零矩阵, 即 A ⊗ O = O ⊗ A = O \boldsymbol A \otimes\boldsymbol O=\boldsymbol O \otimes\boldsymbol A=\boldsymbol O A⊗O=O⊗A=O。
- 若 α \alpha α和 β \beta β为常数, 则 α A ⊗ β B = α β ( A ⊗ B ) \alpha \boldsymbol{A} \otimes \beta \boldsymbol{B}=\alpha \beta(\boldsymbol{A} \otimes \boldsymbol{B}) αA⊗βB=αβ(A⊗B)。
- m \mathit{m} m维与 n \mathit{n} n维两个单位矩阵的 K r o n e c k e r \mathit{Kronecker} Kronecker积为 m n \mathit{m n} mn维单位矩阵, 即 I m ⊗ I n = I m n \boldsymbol{I}_{\mathit{m} } \otimes \boldsymbol{I}_{\mathit{n} }=\boldsymbol{I}_{\mathit{m n} } Im⊗In=Imn。
- 对于矩阵 A m × n , B n × k , C l × p , D p × q \boldsymbol A _{\mathit{m} \times \mathit{n} },\boldsymbol B_{\mathit{n} \times \mathit{k} },\boldsymbol C_{\mathit l \times\mathit p},\boldsymbol D_{\mathit p \times\mathit q} Am×n,Bn×k,Cl×p,Dp×q，有 ( A B ) ⊗ ( C D ) = ( A ⊗ C ) ( B ⊗ D ) (\boldsymbol A\boldsymbol B) \otimes(\boldsymbol C\boldsymbol D)=(\boldsymbol A \otimes\boldsymbol C)(\boldsymbol B \otimes\boldsymbol D) (AB)⊗(CD)=(A⊗C)(B⊗D)
- 对于矩阵 A m × n , B p × q , C p × q \boldsymbol A_{\mathit m \times\mathit n},\boldsymbol B_{\mathit p \times\mathit q},\boldsymbol C_{\mathit p \times\mathit q} Am×n,Bp×q,Cp×q，有 A ⊗ ( B ± C ) = A ⊗ B ± A ⊗ C ( B ± C ) ⊗ A = B ⊗ A ± C ⊗ A \begin{array}{l} \boldsymbol A \otimes(\boldsymbol B \pm\boldsymbol C)=\boldsymbol A \otimes\boldsymbol B \pm\boldsymbol A \otimes\boldsymbol C \\ (\boldsymbol B \pm\boldsymbol C) \otimes\boldsymbol A=\boldsymbol B \otimes\boldsymbol A \pm\boldsymbol C \otimes\boldsymbol A \end{array} A⊗(B±C)=A⊗B±A⊗C(B±C)⊗A=B⊗A±C⊗A
- K r o n e c k e r \mathit{Kronecker} Kronecker积的转置与复共轭转置 ( A ⊗ B ) T = A T ⊗ B T , ( A ⊗ B ) H = A H ⊗ B H (\boldsymbol A \otimes\boldsymbol B)^{\mathrm{T}}=\boldsymbol A^{\mathrm{T}} \otimes\boldsymbol B^{\mathrm{T}}, \quad(\boldsymbol{A} \otimes \boldsymbol{B})^{\mathrm{H}}=\boldsymbol A^{\mathrm{H}} \otimes\boldsymbol B^{\mathrm{H}} (A⊗B)T=AT⊗BT,(A⊗B)H=AH⊗BH
- K r o n e c k e r \mathit{Kronecker} Kronecker积的逆矩阵和广义逆矩阵 ( A ⊗ B ) − 1 = A − 1 ⊗ B − 1 , ( A ⊗ B ) † = A † ⊗ B † (\boldsymbol A \otimes\boldsymbol B)^{-1}=\boldsymbol A^{-1} \otimes\boldsymbol B^{-1}, \quad(\boldsymbol A \otimes\boldsymbol B)^{\dagger}=\boldsymbol A^{\dagger} \otimes\boldsymbol B^{\dagger} (A⊗B)−1=A−1⊗B−1,(A⊗B)†=A†⊗B†
- K r o n e c k e r \mathit{Kronecker} Kronecker积的秩 r a n k ( A ⊗ B ) = r a n k ( A ) r a n k ( B ) \mathit{rank}(\boldsymbol{A} \otimes \boldsymbol{B})=\mathit{rank}(\boldsymbol{A}) \mathit{rank}(\boldsymbol{B}) rank(A⊗B)=rank(A)rank(B)
- K r o n e c k e r \mathit{Kronecker} Kronecker积的行列式 d e t ( A n × n ⊗ B m × m ) = ( d e t A ) m ( d e t B ) n \mathit{det}\left(\boldsymbol A_{\mathit n \times\mathit n} \otimes\boldsymbol B_{\mathit m \times\mathit m}\right)=(\mathit{det}\boldsymbol A)^{\mathit m}(\mathit{det}\boldsymbol B)^{\mathit n} det(An×n⊗Bm×m)=(detA)m(detB)n
- K r o n e c k e r \mathit{Kronecker} Kronecker积的迹 t r ( A ⊗ B ) = t r ( A ) t r ( B ) \mathit{tr}(\boldsymbol{A} \otimes \boldsymbol{B})=\mathit{tr}(\boldsymbol{A}) \mathit{tr}(\boldsymbol{B}) tr(A⊗B)=tr(A)tr(B)
- 对于矩阵 A m × n , B m × n , C p × q , D p × q \boldsymbol{A}_{\mathit m \times\mathit n}, \boldsymbol{B}_{\mathit m \times\mathit n}, \boldsymbol{C}_{\mathit p \times\mathit q},\boldsymbol D_{\mathit p \times\mathit q} Am×n,Bm×n,Cp×q,Dp×q，有 ( A + B ) ⊗ ( C + D ) = A ⊗ C + A ⊗ D + B ⊗ C + B ⊗ D (\boldsymbol A+\boldsymbol B) \otimes(\boldsymbol C+\boldsymbol D)=\boldsymbol A \otimes\boldsymbol C+\boldsymbol A \otimes\boldsymbol D+\boldsymbol B \otimes\boldsymbol C+\boldsymbol B \otimes\boldsymbol D (A+B)⊗(C+D)=A⊗C+A⊗D+B⊗C+B⊗D
- 对于矩阵 A m × n , B p × q , C k × l \boldsymbol{A}_{\mathit m \times\mathit n},\boldsymbol B_{\mathit p \times\mathit q},\boldsymbol C_{\mathit k \times\mathit l} Am×n,Bp×q,Ck×l，有 ( A ⊗ B ) ⊗ C = A ⊗ ( B ⊗ C ) (\boldsymbol A \otimes\boldsymbol B) \otimes\boldsymbol C=\boldsymbol A \otimes(\boldsymbol B \otimes\boldsymbol C) (A⊗B)⊗C=A⊗(B⊗C)
- 对于矩阵 A m × n , B k × l , C p × q , D r × s \boldsymbol{A}_{\mathit m \times\mathit n}, \boldsymbol{B}_{\mathit k \times\mathit l}, \boldsymbol{C}_{\mathit p \times\mathit q},\boldsymbol D_{\mathit r \times\mathit s} Am×n,Bk×l,Cp×q,Dr×s，有 ( A ⊗ B ) ⊗ ( C ⊗ D ) = A ⊗ B ⊗ C ⊗ D (\boldsymbol A \otimes\boldsymbol B) \otimes(\boldsymbol C \otimes\boldsymbol D)=\boldsymbol A \otimes\boldsymbol B \otimes\boldsymbol C \otimes\boldsymbol D (A⊗B)⊗(C⊗D)=A⊗B⊗C⊗D
- 对于矩阵 A m × n , B p × q , C n × r , D q × s , E r × k , F s × l \boldsymbol A_{\mathit m \times\mathit n},\boldsymbol B_{\mathit p \times\mathit q},\boldsymbol C_{\mathit n \times\mathit r},\boldsymbol D_{\mathit q \times\mathit s},\boldsymbol E_{\mathit r \times\mathit k},\boldsymbol F_{\mathit s \times\mathit l} Am×n,Bp×q,Cn×r,Dq×s,Er×k,Fs×l，有 ( A ⊗ B ) ( C ⊗ D ) ( E ⊗ F ) = ( A C E ) ⊗ ( B D F ) (\boldsymbol{A} \otimes \boldsymbol{B})(\boldsymbol{C} \otimes \boldsymbol{D})(\boldsymbol{E} \otimes \boldsymbol{F})=(\boldsymbol{A C E}) \otimes(\boldsymbol{B D F}) (A⊗B)(C⊗D)(E⊗F)=(ACE)⊗(BDF)
- 对于矩阵 A m × n , B p × q \boldsymbol{A}_{\mathit{m} \times\mathit n}, \boldsymbol{B}_{\mathit p \times\mathit q} Am×n,Bp×q，有 exp ( A ⊗ B ) = exp ( A ) ⊗ exp ( B ) \exp (\boldsymbol{A} \otimes \boldsymbol{B})=\exp (\boldsymbol{A}) \otimes\exp (\boldsymbol{B}) exp(A⊗B)=exp(A)⊗exp(B)
文章来源：https://www.toymoban.com/news/detail-729498.html
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