The answer in this question gives an example for the statement
product of two invertible diagonalizable matrices is not diagonalizable.
My question is:
Are there some conditions, perhaps on the eigenvalues and eigenvectors of matrices, under which the product of two invertible diagonalizable matrices is diagonalizable?
If you want to diagonalize a matrix using unitary operators, then it must be normal, i.e.
$A=UDU^\dagger$ if and only if $AA^\dagger = A^\dagger A$.
For your question, you would require that the product is normal:
$AB(AB)^\dagger=(AB)^\dagger AB$,
i.e., $ABB^\dagger A^\dagger = B^\dagger A^\dagger A B$
If two matrices $\;A,B\;$ are simultaneously diagonalizable (and thus they commute, for example), we get $$\begin{cases}A=P^{-1}D_1P\\{}\\B=P^{-1}D_2P\end{cases}\;\;\;\implies AB=P^{-1}D_1D_2P$$
and clearly $\;D_1D_2\;$ is also diagonal.
