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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?

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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. $AB = I$.

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  • $\begingroup$ Why do the eigenvalues have to be distinct? $\endgroup$ – Delphine Martres Mar 18 '19 at 17:19
  • $\begingroup$ 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. $\endgroup$ – Robert Israel Mar 18 '19 at 21:10

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