Product of two symmetric matrices is similar to a symmetric matrix

Question

Let $A,B$ be symmetric real matrices. Is $AB$ similar to a symmetric matrix?

This is a problem in my exam. Not a conjecture :v

Answer 1

Here is a counter-example of two symmetric matrices $A$, $B$ whose product, besides being non symmetrical, cannot be similar to a symmetric matrix.

Consider matrices

$$A=\pmatrix{1&2\\2&1} \ \ \ \text{and} \ \ \ B=\pmatrix{1&0\\0&-1}.$$

$AB=\pmatrix{1&-2\\2&-1}.$ which is non symmetric.

Moreover, the characteristic polynomial of $AB$ is $\lambda^2+3$: thus, the eigenvalues of $AB$ are $\pm i \sqrt{3}$. If it was similar to a symmetric matrix, it would have the same real eigenvalues.