Kronecker Product

Definition

Let $A$ be an $m\times n$ matrix and $B$ be a $p\times q$ matrix, then the Kronecker product $A\otimes B$ is the $pm\times qn$ block matrix $A\otimes B=\left[\begin{array}{ccc}{a}_{11}B& \cdots & a_{1n}B\\ ⋮& \ddots & ⋮\\ a_{m1}B& \cdots & a_{mn}B\end{array}\right]$

Facts

Let $A,B,X,Y$ be matrices, then

$(A\otimes B{)}^{⊺}=A^{⊺}\otimes B^{⊺}$ $(A\otimes B)(X\otimes Y)=AX\otimes BY$ $(A\otimes B{)}^{-1}=A^{-1}\otimes B^{-1}$, where $A,B$ are square matrices $\mathrm{rank}(A\otimes B)=\mathrm{rank}(A)\mathrm{rank}(B)$ $\mathrm{tr}(A\otimes B)=\mathrm{tr}(A)\mathrm{tr}(B)$ $|A_{p\times p}\otimes B_{m\times m}|=|A{|}^m|B{|}^p$ Eigenvalues of $A\otimes B$ is the product of the eigenvalue of $A$ and the eigenvalue of $B$