# Are the eigenvalues of the product matrix of two real symmetric square matrices also real values?

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?

• 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. – Daniel Shapero Feb 7 '14 at 16:32
• This question on the math.SE site about the eigenvalues of a product of SPD matrices may be of interest to you. – Paul Feb 8 '14 at 14:45