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$\mathbf{A}\in {\mathbb{R}}^{m\times n}$和$\mathbf{B}\in {\mathbb{R}}^{p\times q}$，则这两个矩阵的克罗内克积是一个在空间${\mathbb{R}}^{mp\times nq}$的分块矩阵$$\mathbf{A}\otimes \mathbf{B}=\left[\begin{array}{ccc}{a}_{11}\mathbf{B}& \cdots & a_{1n}\mathbf{B}\\ ⋮& \ddots & ⋮\\ a_{m1}\mathbf{B}& \cdots & a_{mn}\mathbf{B}\end{array}\right]$$更具体的可以表示为$$\mathbf{A}\otimes \mathbf{B}=\left[\begin{array}{cccccccccc}{a}_{11}{b}_{11}& a_{11}{b}_{12}& \cdots & a_{11}{b}_{1q}& \cdots & \cdots & a_{1n}{b}_{11}& a_{1n}{b}_{12}& \cdots & a_{1n}{b}_{1q}\\ a_{11}{b}_{21}& a_{11}{b}_{22}& \cdots & a_{11}{b}_{2q}& \cdots & \cdots & a_{1n}{b}_{21}& a_{1n}{b}_{22}& \cdots & a_{1n}{b}_{2q}\\ ⋮& ⋮& \ddots & ⋮& & & ⋮& ⋮& \ddots & ⋮\\ a_{11}{b}_{p1}& a_{11}{b}_{p2}& \cdots & a_{11}{b}_{pq}& \cdots & \cdots & a_{1n}{b}_{p1}& a_{1n}{b}_{p2}& \cdots & a_{1n}{b}_{pq}\\ ⋮& ⋮& & ⋮& \ddots & & ⋮& ⋮& & ⋮\\ ⋮& ⋮& & ⋮& & \ddots & ⋮& ⋮& & ⋮\\ a_{m1}{b}_{11}& a_{m1}{b}_{12}& \cdots & a_{m1}{b}_{1q}& \cdots & \cdots & a_{mn}{b}_{11}& a_{mn}{b}_{12}& \cdots & a_{mn}{b}_{1q}\\ a_{m1}{b}_{21}& a_{m1}{b}_{22}& \cdots & a_{m1}{b}_{2q}& \cdots & \cdots & a_{mn}{b}_{21}& a_{mn}{b}_{22}& \cdots & a_{mn}{b}_{2q}\\ ⋮& ⋮& \ddots & ⋮& & & ⋮& ⋮& \ddots & ⋮\\ a_{m1}{b}_{p1}& a_{m1}{b}_{p2}& \cdots & a_{m1}{b}_{pq}& \cdots & \cdots & a_{mn}{b}_{p1}& a_{mn}{b}_{p2}& \cdots & a_{mn}{b}_{pq}\end{array}\right]$$

克罗内克积性质

 克罗内克积满足如下性质：

• 结合律：$(\mathbf{A}\otimes \mathbf{B})\otimes \mathbf{C}=\mathbf{A}\otimes (\mathbf{B}\otimes \mathbf{C})$
• 分配律：$\mathbf{A}\otimes (\mathbf{B}+\mathbf{C})=\mathbf{A}\otimes \mathbf{B}+\mathbf{A}\otimes \mathbf{C}$ $(\mathbf{A}+\mathbf{B})\otimes \mathbf{C}=\mathbf{A}\otimes \mathbf{C}+\mathbf{B}\otimes \mathbf{C}$
• 双线性：$(k\mathbf{A})\otimes \mathbf{B}=\mathbf{A}\otimes (k\mathbf{B})=k(\mathbf{A}\otimes \mathbf{B})$
• 转置：$(\mathbf{A}\otimes \mathbf{B}{)}^{\top }={\mathbf{A}}^{\top }\otimes {\mathbf{B}}^{\top }$
• 混合乘积性：如果$\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$是四个矩阵，且矩阵乘积$\mathbf{A}\mathbf{C}$和$\mathbf{B}\mathbf{D}$存在，那么$(\mathbf{A}\otimes \mathbf{B})(\mathbf{C}\otimes \mathbf{D})=\mathbf{A}\mathbf{C}\otimes \mathbf{B}\mathbf{D}$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，$\mathbf{A}\otimes \mathbf{B}$是可逆的当且仅当$\mathbf{A}$和$\mathbf{B}$是可逆的，其逆矩阵为$$(\mathbf{A}\otimes \mathbf{B}{)}^{-1}={\mathbf{A}}^{-1}\otimes {\mathbf{B}}^{-1}$$
• $vec({\mathbf{a}\mathbf{b}}^{\top })=\mathbf{b}\otimes \mathbf{a}$

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

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