Kronecker积
Kronecker积（也称为张量积）是一种矩阵运算，用于将两个矩阵组合成一个更大的块矩阵。
- 如果矩阵 A\bm AA 的维度为 m×nm \times nm×n，矩阵 B\bm BB 的维度为 p×qp \times qp×q，那么它们的Kronecker积 A⊗B\bm A \otimes \bm BA⊗B 是一个 mp×nqmp \times nqmp×nq 矩阵。
- 运算规则：A⊗B\bm A \otimes \bm BA⊗B 的第 (i,j)(i, j)(i,j) 个块是 aijBa_{ij} \bm BaijB，即 A\bm AA 的每个元素乘以整个 B\bm BB 矩阵。
例如：
A=[a11a12a21a22],B=[b11b12b21b22] \boldsymbol{A}=\left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} b_{11} & b_{12} \\ b_{21} & b_{22} \end{array}\right] A=[a11a21a12a22],B=[b11b21b12b22]
则：
A⊗B=[a11Ba12Ba21Ba22B]=[a11b11a11b12a12b11a12b12a11b21a11b22a12b21a12b22a21b11a21b12a22b11a22b12a21b21a21b22a22b21a22b22] \boldsymbol{A} \otimes \boldsymbol{B}=\left[\begin{array}{cc} a_{11} \boldsymbol{B} & a_{12} \boldsymbol{B} \\ a_{21} \boldsymbol{B} & a_{22} \boldsymbol{B} \end{array}\right]=\left[\begin{array}{cccc} a_{11} b_{11} & a_{11} b_{12} & a_{12} b_{11} & a_{12} b_{12} \\ a_{11} b_{21} & a_{11} b_{22} & a_{12} b_{21} & a_{12} b_{22} \\ a_{21} b_{11} & a_{21} b_{12} & a_{22} b_{11} & a_{22} b_{12} \\ a_{21} b_{21} & a_{21} b_{22} & a_{22} b_{21} & a_{22} b_{22} \end{array}\right] A⊗B=[a11Ba21Ba12Ba22B]=a11b11a11b21a21b11a21b21a11b12a11b22a21b12a21b22a12b11a12b21a22b11a22b21a12b12a12b22a22b12a22b22