Real Matrices with Real Eigenvalue pre- and Post multiplied by a Diagonal Matrix

Question

Suppose all the eigenvalues of $A\in \mathbb{R}^{n\times n}$ (not necessarily symmetric) are real. Let $D\in \mathbb{R}^{n\times n}$ be a diagonal matrix with positive diagonals. Prove/disprove that $A+D$ and $DAD$ has only real eigenvalues.

Answer 1

I played around with Maple and came up with a counterexample. I'm not going to prove it's a counterexample as the mathematics is tedious.

Take $$A = \begin{pmatrix} 1 & 3 & 2 \\ -1 & 1 & 4 \\ 1 & 2 & 7 \end{pmatrix}^{-1}\begin{pmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 3 \end{pmatrix}\begin{pmatrix} 1 & 3 & 2 \\ -1 & 1 & 4 \\ 1 & 2 & 7 \end{pmatrix},$$ and $$D = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1\end{pmatrix}.$$ Then $DAD$ and $A + D$ has non-real eigenvalues.