Kronecker product

Kronecker product 的基本运算

结合律

\begin{equation}

\mathrm{A} \otimes (\mathrm{B + C}) = \mathrm{A} \otimes \mathrm{B} + \mathrm{A}\otimes \mathrm{C}

\end{equation}

\begin{equation}

(\mathrm{A} + \mathrm{B} ) \otimes \mathrm{C} = \mathrm{A} \otimes \mathrm{C} + \mathrm{B} \otimes \mathrm{C}

\end{equation}

转置运算

\begin{equation}

(\mathrm{A} \otimes \mathrm{B})^{T} = \mathrm{A}^{T} \otimes \mathrm{B}^{T}

\end{equation}

分配率

\begin{equation}

(\mathrm{A} \otimes \mathrm{B})(\mathrm{C} \otimes \mathrm{D}) = \mathrm{AC} \otimes \mathrm{BD}

\end{equation}

逆运算

\begin{equation}

(\mathrm{A} \otimes \mathrm{B})^{-1} = \mathrm{A}^{-1} \otimes \mathrm{B}^{-1}

\end{equation}

Det运算：

\begin{equation}

\left|\mathbf{I}_{T} \otimes \Sigma\right|=|\Sigma|^{T}

\end{equation}

与vector相结合

参考论文：Large Bayesian Vector Autoregressions

\[ \begin{align}
\operatorname{vec}(\mathbf{B C D}) &=\left(\mathbf{D}^{\prime} \otimes \mathbf{B}\right) \operatorname{vec}(\mathbf{C}) \\
\operatorname{tr}\left(\mathbf{B}^{\prime} \mathbf{C}\right) &=\operatorname{vec}(\mathbf{B})^{\prime} \operatorname{vec}(\mathbf{C}), \\
\operatorname{tr}(\mathbf{B C D}) &=\operatorname{tr}(\mathbf{C D B})=\operatorname{tr}(\mathbf{D B C}),
\end{align} \]