[Math] Eigenvalues of the Product of a Diagonal and a Symmetric Matrix

Question

Let $A \in \mathbb{R}^{n\times n}$ be a symmetric matrix and $D \in \mathbb{R}^{n\times n}$ be a diagonal matrix with positive entries. Prove that the matrix $P:=DA$ has real eigenvalues.

Answer 1

The characteristic equation of matrix $P$ is \begin{align} \text{det}(\lambda I - DA) = \text{det}( D^{\frac{1}{2}} (\lambda I - D^{\frac{1}{2}}AD^{\frac{1}{2}}) D^{\frac{-1}{2}}) =0 \end{align} Thus the eigenvalues of $P=DA$ are the same as the eigenvalues of $Q:=D^{\frac{1}{2}}AD^{\frac{1}{2}}$ which is symmetric and has real eigenvalues.