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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?

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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.

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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., $ABB^\dagger A^\dagger = B^\dagger A^\dagger A B$

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