Spectrum of product from two selfadjoint matricies

Question

If I have 2 selfadjoint matricies A,B given, is the spectrum of

$A B$

always real? I know that $A B$ is not necessary a selfadjoint matrix, but are some properties of the spectrum preserved?

Answer 1

I believe this is the counterexample you are looking for: $$\begin{bmatrix}1&0\\0&-1\end{bmatrix} \begin{bmatrix}0&1\\1&0\end{bmatrix}=\begin{bmatrix}0&1\\-1&0\end{bmatrix}$$ As the characteristic polynomial of the product is $\lambda^2 + 1$ and its eigenvalues are therefore $i$ and $-i$.