Eigenvalues of A^T D A for positive A and diagonal D

Question

Suppose I have a diagonal matrix $D$ whose entries are bounded in absolute value. I also have a matrix $A$ that is positive (entry-wise, so $A_{ij} > 0\ \forall\ i,j$): one can assume that the entries of $A$ are all bounded away from 0 by $a_0$. I would like to understand what the eigenvalues of the matrix $A^T D A$ look like, and in particular how they compare to the eigenvalues of $D$. Can one say anything about $x^TA^T D Ax$ compared to $x^T D x$?

EDIT: If it helps, one can also assume the $A$ is PD.

Answer 1

The eigenvalues $\sigma_i$ of $A^T DA$ are the same as those of $MD$ with $M=AA^{T}$ a positive definite matrix. Denote the eigenvalues of $M$ by $\mu_i$ and the diagonal elements of $D$ by $\delta_i$. Let me assume that the $\mu_i$'s and $\delta_i$'s are all positive. Each set of $n$ eigenvalues is ordered from large to small, $\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_n>0$, and similarly for the $\mu_i$ and $\delta_i$. Then Horn's inequalities give $$\sigma_{i+j-1}\leq \mu_i\delta_j\leq\sigma_{i+j-n}.$$