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Suppose $A,B \in \mathbb{R}^{n\times n}; A=A^T, B=B^T$, let $C = AB, D =BA$,

If we have all the real eigenvalues of $A$ and $B$, e.g. the eigenvalue decomposition of them:

$A=P\Lambda_1 P^T$, $B=Q\Lambda_2 Q^T$; both $P$ and $Q$ are orthogonal matrices, and both $\Lambda_1$ and $ \Lambda_2$ are diagnoal matrices;

can we also obtain the eigenvalues of $C$ and $D$ based on these information without solving eigenvalue problems of $C$ and $D$?

Are the eigenvalues of $C$ and $D$ also real?

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    $\begingroup$ Usually you can only do this if you know that $A$ and $B$ commute, i.e. $AB = BA$. In that case, $A$ and $B$ have the same eigenvectors. $\endgroup$
    – Daniel Shapero
    Feb 7, 2014 at 16:32
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    $\begingroup$ This question on the math.SE site about the eigenvalues of a product of SPD matrices may be of interest to you. $\endgroup$
    – Paul
    Feb 8, 2014 at 14:45

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