# Simultaneous diagonalization of matrices by using their product

If $$A$$ and $$B$$ are two simultaneously diagonalizable normal matrices, is it possible to find the common unitary matrix $$U$$ formed by their common eigenvectors by diagonalizing their product $$AB$$ since it is also diagonalizable by the same unitary matrix?

Presumably you're talking about normal matrices, otherwise there's no reason for $$U$$ to be unitary.
It will be possible if $$AB$$ has distinct eigenvalues. On the other hand, you might have e.g. $$AB = I$$.
• If they are not, you have lots of ways to diagonalize the matrix $AB$ which might not diagonalize $A$ or $B$. A worst case scenario is $AB = I$: then any invertible matrix $U$ can be used to diagonalize it. – Robert Israel Mar 18 '19 at 21:10