[Math] Eigenvalues of a symmetric positive definite matrix multiplied by a diagonal matrix

Question

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.)

Answer 1

You do have in general inequalities for singular values of the product of two matrices $M$, $N$, due to Weyl or Schur, I forgot, Horn and Johnson should have the reference. If $\alpha_i$, $\beta_i$ $\gamma_i$ are the singular values of $M$, $N$ and $MN$, in decreasing order then $$\alpha_1 \cdot \beta_1\ge \gamma_1 \\ \alpha_1 \alpha_2 \ge \gamma_1 \gamma_2\ge \gamma_1 \gamma_2\\ \ldots\ldots\ldots\\ \alpha_1 \cdots\alpha_n \cdot\beta_1 \cdots \beta_n = \gamma_1 \cdots \gamma_n$$ The first inequality is not hard, since it is the inequality for the $l^2$ norm. The other are obtained considering the associated operators $\wedge^k M, N, MN$.

So, fixing $\lambda_i$ for a hermitian, and the $d_i$ there definitely are inequalities for the eigenvalues of $DAD$. Hard to tell what are all the defining inequalities.