Diagonalisation and Kronecker Product If $A$ is a $n\times n$ matrix with complex numbers for elements, and $C$ the $2\times2$ matrix defined by 
$$\begin{bmatrix}
-2&amp;4\\-3&amp;5
\end{bmatrix}.$$
How do you prove that the Kronecker product $C\otimes A$ is diagonalisable if and only if $A$ is diagonalisable?
Use the properties of the Kronecker product:
$$
(P^{-1}\otimes Q^{-1})(C \otimes A) (P \otimes Q)
= (P^{-1}\otimes Q^{-1})( CP\otimes AQ) = (P^{-1}CP)\otimes (Q^{-1}AQ)
$$
The given matrix $C$ is diagonalizable (it has two different eigenvalues), thus, there exists $P$ such that $P^{-1}CP$ is diagonal.

If $A$ is diagonalizable, then clearly $C\otimes A$ is diagonalizable.
This claim is not correct. See exercise 15 in chapter 4.3 in Horn &amp; Johnson's Topics in Matrix Analysis, which states that

$C \otimes A$ is diagonalisable if and only if both $A$ and $C$ is diagonalisable.