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

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.

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.