克罗内克积（Kronecker）

鬼道2022

40200人浏览 · 2021-04-16 16:56:43

鬼道2022 · 2021-04-16 16:56:43 发布

克罗内克积定义

克罗内克积是两个任意大小的矩阵间的运算，它是张量积的特殊形式。给定两个矩阵 $A∈Rm×n\boldsymbol{A} \in \mathbb{R}^{m \times n}$ 和 $B∈Rp×q\boldsymbol{B} \in \mathbb{R}^{p \times q}$ ，则这两个矩阵的克罗内克积是一个在空间 $Rmp×nq\mathbb{R}^{mp \times nq}$ 的分块矩阵 $A⊗B=[a11B⋯a1nB⋮⋱⋮am1B⋯amnB]\boldsymbol{A}\otimes\boldsymbol{B}=\left[\begin{array}{ccc}a_{11}\boldsymbol{B}&\cdots & a_{1n}\boldsymbol{B}\\\vdots&\ddots&\vdots\\a_{m1}\boldsymbol{B}&\cdots&a_{mn}\boldsymbol{B}\end{array}\right]$ 更具体的可以表示为 $\boldsymbol{A} \otimes \boldsymbol{B}=\left[\begin{array}{ccccccccc} a_{11} b_{11} & a_{11} b_{12} & \cdots & a_{11} b_{1 q} & \cdots & \cdots & a_{1 n} b_{11} & a_{1 n} b_{12} & \cdots & a_{1 n} b_{1 q} \\ a_{11} b_{21} & a_{11} b_{22} & \cdots & a_{11} b_{2 q} & \cdots & \cdots & a_{1 n} b_{21} & a_{1 n} b_{22} & \cdots & a_{1 n} b_{2 q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{11} b_{p 1} & a_{11} b_{p 2} & \cdots & a_{11} b_{p q} & \cdots & \cdots & a_{1 n} b_{p 1} & a_{1 n} b_{p 2} & \cdots & a_{1 n} b_{p q} \\ \vdots & \vdots & & \vdots & \ddots & & \vdots & \vdots & & \vdots \\ \vdots & \vdots & & \vdots & & \ddots & \vdots & \vdots & & \vdots \\ a_{m 1} b_{11} & a_{m 1} b_{12} & \cdots & a_{m 1} b_{1 q} & \cdots & \cdots & a_{m n} b_{11} & a_{m n} b_{12} & \cdots & a_{m n} b_{1 q} \\ a_{m 1} b_{21} & a_{m 1} b_{22} & \cdots & a_{m 1} b_{2 q} & \cdots & \cdots & a_{m n} b_{21} & a_{m n} b_{22} & \cdots & a_{m n} b_{2 q} \\ \vdots & \vdots & \ddots & \vdots & & & \vdots & \vdots & \ddots & \vdots \\ a_{m 1} b_{p 1} & a_{m 1} b_{p 2} & \cdots & a_{m 1} b_{p q} & \cdots & \cdots & a_{m n} b_{p 1} & a_{m n} b_{p 2} & \cdots & a_{m n} b_{p q} \end{array}\right]$

克罗内克积性质

克罗内克积满足如下性质：

结合律： $(A⊗B)⊗C=A⊗(B⊗C)(\boldsymbol{A}\otimes\boldsymbol{B})\otimes\boldsymbol{C}=\boldsymbol{A}\otimes(\boldsymbol{B}\otimes\boldsymbol{C})$
分配律： $A⊗(B+C)=A⊗B+A⊗C\boldsymbol{A}\otimes(\boldsymbol{B}+\boldsymbol{C})=\boldsymbol{A}\otimes \boldsymbol{B}+\boldsymbol{A}\otimes\boldsymbol{C}\quad\quad$ $(A+B)⊗C=A⊗C+B⊗C(\boldsymbol{A}+\boldsymbol{B})\otimes\boldsymbol{C}=\boldsymbol{A}\otimes\boldsymbol{C}+\boldsymbol{B}\otimes\boldsymbol{C}$
双线性： $(kA)⊗B=A⊗(kB)=k(A⊗B)(k\boldsymbol{A})\otimes\boldsymbol{B}=\boldsymbol{A}\otimes(k\boldsymbol{B})=k(\boldsymbol{A}\otimes\boldsymbol{B})$
转置： $(A⊗B)⊤=A⊤⊗B⊤(\boldsymbol{A} \otimes \boldsymbol{B})^{\top}=\boldsymbol{A}^{\top} \otimes \boldsymbol{B}^{\top}$
混合乘积性：如果 $A,B,C,D\boldsymbol{A,B,C,D}$ 是四个矩阵，且矩阵乘积 $AC\boldsymbol{AC}$ 和 $BD\boldsymbol{BD}$ 存在，那么 $(A⊗B)(C⊗D)=AC⊗BD(\boldsymbol{A}\otimes\boldsymbol{B})(\boldsymbol{C}\otimes\boldsymbol{D})=\boldsymbol{AC}\otimes\boldsymbol{BD}$ 这个性质称为“混合乘积性质”，因为它混合了矩阵乘积和克罗内克积。可以推出， $A⊗B\boldsymbol{A}\otimes\boldsymbol{B}$ 是可逆的当且仅当 $A\boldsymbol{A}$ 和 $B\boldsymbol{B}$ 是可逆的，其逆矩阵为 $(A⊗B)−1=A−1⊗B−1(\boldsymbol{A}\otimes\boldsymbol{B})^{-1}=\boldsymbol{A}^{-1}\otimes\boldsymbol{B}^{-1}$
$vec(ab⊤)=b⊗avec(\bm{ab}^{\top})=\bm{b} \otimes \bm{a}$

克罗内克积实例

实例1

$\left[\begin{array}{ll} 1 & 2 \\ 3 & 1 \end{array}\right] \otimes\left[\begin{array}{ll} 0 & 3 \\ 2 & 1 \end{array}\right]=\left[\begin{array}{llll} 1 \cdot 0 & 1 \cdot 3 & 2 \cdot 0 & 2 \cdot 3 \\ 1 \cdot 2 & 1 \cdot 1 & 2 \cdot 2 & 2 \cdot 1 \\ 3 \cdot 0 & 3 \cdot 3 & 1 \cdot 0 & 1 \cdot 3 \\ 3 \cdot 2 & 3 \cdot 1 & 1 \cdot 2 & 1 \cdot 1 \end{array}\right]=\left[\begin{array}{llll} 0 & 3 & 0 & 6 \\ 2 & 1 & 4 & 2 \\ 0 & 9 & 0 & 3 \\ 6 & 3 & 2 & 1 \end{array}\right]$

实例2

$\left[\begin{array}{l} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right] \otimes\left[\begin{array}{lll} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{array}\right]=\left[\begin{array}{llllll} a_{11} b_{11} & a_{11} b_{12} & a_{11} b_{13} & a_{12} b_{11} & a_{12} b_{12} & a_{12} b_{13} \\ a_{11} b_{21} & a_{11} b_{22} & a_{11} b_{23} & a_{12} b_{21} & a_{12} b_{22} & a_{12} b_{23} \\ a_{21} b_{11} & a_{21} b_{12} & a_{21} b_{13} & a_{22} b_{11} & a_{22} b_{12} & a_{22} b_{13} \\ a_{21} b_{21} & a_{21} b_{22} & a_{21} b_{23} & a_{22} b_{21} & a_{22} b_{22} & a_{22} b_{23} \\ a_{31} b_{11} & a_{31} b_{12} & a_{31} b_{13} & a_{32} b_{11} & a_{32} b_{12} & a_{32} b_{13} \\ a_{31} b_{21} & a_{31} b_{22} & a_{31} b_{23} & a_{32} b_{21} & a_{32} b_{22} & a_{32} b_{23} \end{array}\right]$