克罗内克积是线性代数中的一种矩阵运算，将两个矩阵组合成一个大矩阵。本文介绍了如何计算两个矩阵的克罗内克积，并列举了其重要的性质，如分配律和结合律等。克罗内克积在数学和工程领域中有广泛应用。
克罗内克积[Kronecker product]
已知矩阵A和矩阵B。
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\textbf{A}_{mn}=\begin{bmatrix} a_{11} & a_{12} & ... & a_{1n} \\ a_{21} & a_{22} & ... & a_{2n} \\ ... & ... & ... & ... \\ a_{m1} & a_{m2} & ... & a_{mn} \\ \end{bmatrix},\textbf{B}_{pq}=\begin{bmatrix} b_{11} & b_{12} & ... & b_{1q} \\ b_{21} & b_{22} & ... & b_{2q} \\ ... & ... & ... & ... \\ b_{p1} & b_{p2} & ... & b_{pq} \\ \end{bmatrix}
Amn=⎣
⎡a11a21...am1a12a22...am2............a1na2n...amn⎦
⎤,Bpq=⎣
⎡b11b21...bp1b12b22...bp2............b1qb2q...bpq⎦
⎤
进行克罗内克积运算
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\textbf{A}\otimes \textbf{B}=\begin{bmatrix} a_{11}\textbf{B} & a_{12}\textbf{B} & ... & a_{1n}\textbf{B} \\ a_{21}\textbf{B} & a_{22}\textbf{B} & ... & a_{2n}\textbf{B} \\ ... & ... & ... & ... & \\ a_{m1}\textbf{B} & a_{m2}\textbf{B} & ... & a_{mn}\textbf{B} \\ \end{bmatrix}=\\ \begin{bmatrix} a_{11}b_{11} & ... & a_{11}b_{1q} & a_{12}b_{11} & ... & a_{12}b_{1q} & ... & a_{1n}b_{11} & ... & a_{1n}b_{1q} \\ ... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\ a_{11}b_{p1} & ... & a_{11}b_{pq} & a_{12}b_{p1} & ... & a_{12}b_{pq} & ... & a_{1n}b_{p1} & ... & a_{1n}b_{pq} \\ a_{21}b_{11} & ... & a_{21}b_{1q} & a_{22}b_{11} & ... & a_{22}b_{1q} & ... & a_{2n}b_{11} & ... & a_{2n}b_{1q} \\ ... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\ a_{21}b_{p1} & ... & a_{21}b_{pq} & a_{22}b_{p1} & ... & a_{22}b_{pq} & ... & a_{2n}b_{p1} & ... & a_{2n}b_{pq} \\ ... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\ a_{m1}b_{11} & ... & a_{m1}b_{1q} & a_{m2}b_{11} & ... & a_{m2}b_{1q} & ... & a_{mn}b_{11} & ... & a_{mn}b_{1q} \\ ... & ... & ... & ... & ... & ... & ... & ... & ... & ... \\ a_{m1}b_{p1} & ... & a_{m1}b_{pq} & a_{m2}b_{p1} & ... & a_{m2}b_{pq} & ... & a_{mn}b_{p1} & ... & a_{mn}b_{pq} \\ \end{bmatrix}
A⊗B=⎣
⎡a11Ba21B...am1Ba12Ba22B...am2B............a1nBa2nB...amnB⎦
⎤=⎣
⎡a11b11...a11bp1a21b11...a21bp1...am1b11...am1bp1..............................a11b1q...a11bpqa21b1q...a21bpq...am1b1q...am1bpqa12b11...a12bp1a22b11...a22bp1...am2b11...am2bp1..............................a12b1q...a12bpqa22b1q...a22bpq...am2b1q...am2bpq..............................a1nb11...a1nbp1a2nb11...a2nbp1...amnb11...amnbp1..............................a1nb1q...a1nbpqa2nb1q...a2nbpq...amnb1q...amnbpq⎦
⎤
性质[properties]
Kronecker product是张量积 [tensor product] 的一种特殊形式，具有如下性质：
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\textbf{A}\otimes (\textbf{B}+\textbf{C})=\textbf{A}\otimes \textbf{B}+\textbf{A}\otimes \textbf{C},\\ (\textbf{B}+\textbf{C})\otimes \textbf{A}=\textbf{B}\otimes \textbf{A}+\textbf{C}\otimes \textbf{A},\\ (k\textbf{A})\otimes\textbf{B}=k\textbf{A}\otimes \textbf{B},\\ (\textbf{A}\otimes\textbf{B})\otimes\textbf{C}=\textbf{A}\otimes(\textbf{B}\otimes\textbf{C})
A⊗(B+C)=A⊗B+A⊗C,(B+C)⊗A=B⊗A+C⊗A,(kA)⊗B=kA⊗B,(A⊗B)⊗C=A⊗(B⊗C)
【注：还有其他性质，可参考Wikipedia中的详述，如果后续应用过程中遇到相关性质再进行补充】
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