# Eigenvalues of a symmetric positive definite matrix multiplied by a diagonal matrix

Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and $D = \text{diag} (d_1, d_2, ... , d_n)$ be a positive diagonal matrix. We know that eigenvalues of A are $\lambda_1 \leq \lambda_2 \leq ... \leq \lambda_n$. What will be eigenvalues of $DAD$? (Obviously they're all positive.)

• You don't mean $DAD^{-1}$, right? Similar matrices have the same eigenvalues (math.stackexchange.com/questions/8339/…) – parsiad Aug 8 '15 at 3:21
• Absolutly, I meant $DAD$. You're right about similar matrices though. – questioner Aug 8 '15 at 4:09
• It's impossible to give an exact result without more information about $A$. However, we might be able to get some useful inequalities. – Omnomnomnom Aug 8 '15 at 4:28
• Assume $A$ is laplacian of an undirected connected graph. (I know $A$ is positive semi-definite in this case). What inequalities? (beside the most obvious ones) – questioner Aug 8 '15 at 4:32
• I'd have to look through Bhatia's matrix analysis to see if something applies; I can't think of anything relevant off the top of my head. – Omnomnomnom Aug 8 '15 at 12:37