Question

Suppose that we have a symmetric matrix ${\bf S}$ with eigenvalue decomposition ${\bf S} = {\bf Q}{\bf \Lambda}{\bf Q}^T$. Assume that we have two diagonal matrices ${\bf D}_1$ and ${\bf D}_2$ that are multiplying ${\bf S}$ from the left and right, i.e. ${\bf A} = {\bf D}_1{\bf S}{\bf D}_2$. Can we relate the eigenvalues of ${\bf S}$ to the ones of ${\bf A}$? How about the case where ${\bf S}$ is not symmetric?

Answer 1

In general there is no relation: for example consider the simplest case $S$ itself is diagonal and invertible. Letting $D_1=S^{-1}$ then $A$ can be any diagonal matrix $D_2$. The only considerations you can do are related to the presence of the zero eigenvalues using Binet formula for determinants. Notice also that in general $A$ itself can be nonsymmetric, and its eigenvalues can be complex. However small perturbations, i.e. small $D_1$ and $D_2$, result in a small perturbation of the eigenvalues of $S$ in the complex plane.