When is the product of symmetric matrices symmetric?

Question

Suppose I have two symmetric matrices $A, B\in\mathbb{R}^{n\times n}$. What qualifiers do I need to ensure that $B^{-1}AB$ is also symmetric? Is $B$ being diagonal enough?

Answer 1

Since $B^{-1}AB=B^{-1T}A^TB^T=(BAB^{-1})^T$, $B^{-1}AB-(B^{-1}AB)^T=(BAB^{-1}-B^{-1}AB)^T$, so the desired condition is equivalent to $BAB^{-1}=B^{-1}AB$ i.e. $[A,\,B^2]=O$, which occurs if $A,\,B^2$ are simultaneously diagonalizable. Clearly, however, not all diagonal $B$ satisfy $[A,\,B^2]=O$, even if $A$ is symmetric.