Product of two symmetric banded matrices - real eigenvalues? Let $A$ and $B$ be real symmetric banded matrices but $AB$ is not symmetric.  Are the eigenvalues of $AB$ real?
A more specific case: let $D$ be a real diagonal matrix, $B$ real symmetric and banded, and $DB$ is not symmetric.  Are the eigenvalues of $DB$ real?

No, 
$
\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}
\begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}=
\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}
$.
