Basic property of kronecker delta $( A \otimes B)(A^{-1} \otimes B^{-1})$

Question

Given non singular matrices $A_{n \times n},B_{m \times m}$

$$ ( A \otimes B)(A^{-1} \otimes B^{-1}) = (AA^{-1}) \otimes (BB^{-1}) = I_n \otimes I_m = I_{(nm \times nm )} $$

I was just reading through mathematical primer for social statistics by John Fox and saw this on page 17, it wasn't clear to me why this is true though.

Answer 1

The basic idea is that $$A\otimes B =\begin{bmatrix} a_{11}B & a_{12}B & a_{13}B \dots \\ a_{21}B & a_{22}B & a_{23}B \dots \\ \vdots & \vdots & \vdots \end{bmatrix} $$

$$A^{-1}\otimes B^{-1} =\begin{bmatrix} a_{11}'B^{-1} & a_{12}'B^{-1} & a_{13}'B^{-1} \dots \\ a_{21}'B^{-1} & a_{22}'B^{-1} & a_{23}'B^{-1} \dots \\ \vdots & \vdots & \vdots \end{bmatrix} $$

$$(A\otimes B)(A^{-1}\otimes B^{-1}) =\begin{bmatrix} c_{11}BB^{-1} & c_{12}BB^{-1} & c_{13}BB^{-1} \dots \\ c_{21}BB^{-1} & c_{22}BB^{-1} & c_{23}BB^{-1} \dots \\ \vdots & \vdots & \vdots \end{bmatrix} = \begin{bmatrix} c_{11}I & c_{12}I & c_{13}I \dots \\ c_{21}I & c_{22}I & c_{23}I \dots \\ \vdots & \vdots & \vdots \end{bmatrix}$$

where $c_{ij}$ is a an element of $AA^{-1}$, so $c_{ij} = \delta_{ij}$ and the equation holds. This is extremely schematic of course, just to give you some "visuals". You should calculate $A\otimes B$, $A^{-1}\otimes B^{-1}$ directly by definition of Kronecker product and then multiply them to see how indices behave.

Answer 2

More generally, if $A,B,C,D$ are matrices such that the products $AC$ and $BD$ are defined, then $(A\otimes B)(C\otimes D)=AC\otimes BD$. This follows from the definition of Kronecker product.

Another relevant property here is $I_m\otimes I_n=I_{mn}$.