$A$, $B$ are two matrices such that $A\ge0$ and $B\ge0$ and either $A>0$ or $B>0$. I am trying to show that matrix $BA$ is similar to a matrix with non-negative diagonal elements. Here; $A$ and $B$ are symmetric matrices. Let's suppose the two matrices have the same size.
My Approach: \begin{align} BA = B^{1/2}(B^{1/2}AB^{1/2})B^{-1/2} && (i) \end{align}
This is analogous to Schur decomposition; $X = SDS^{-1}$. If $B = SDS^{-1}$ where $D$ is a diagonal matrix and $S$ is an orthogonal matrix; then (i) becomes:
$BA = (SD^{1/2}S^{-1})[(SD^{1/2}S^{-1})A(SD^{1/2}S^{-1})](SD^{-1/2}S^{-1})$
Now, if I could show that $[(SD^{1/2}S^{-1})A(SD^{1/2}S^{-1})]$ has upper-block triangular then I think it would help me answer the question. Any suggestions?