Product of two symmetric banded matrices - real eigenvalues?

Question

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?

Answer 1

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} $.