Let $A\in \mathbb{R}^{n\times n}$. Moreover, assume $D$ is an $n \times n$ diagonal matrix with positive diagonals. What is the relation between the eigenvalues of $A$ and eigenvalues of $B:=DAD$? In other words, how do the eigenvalues of the matrix $A$ change when it is pre and post multiplied by a diagonal matrix? Do $A$ and $B$ have the same inertia? We can assume that $A$ is diagonalizable if necessary.
Any comment/response is greatly appreciated.
Does congruence preserve inertia only for symmetric $A$? I have a similar problem with blocked $A = \begin{bmatrix}L& B\\B& L\end{bmatrix}$ and $D = \begin{bmatrix}D_1& \\& D_2\end{bmatrix}$; where $L$ is symmetric but $B$ is not symmetric. I know $A$ has all nonnegative eigenvalues, does that hold for $DAD$?