# Bound on the eigenvalues of product of a diagonal matrix and a symmetric matrix

I saw several questions similar to what I am asking but couldn't find what I am looking for. So here it is. Let:

$$J = DA$$

where $D$ is a diagonal matrix and $A$ is a symmetric matrix. I know the eigenvalues of $A$ and I want the bounds on the eigenvalues of $J$ in terms of eigenvalues of $A$ and $D$. Here, $D$ has all positive eigenvalues (the elements on its diagonal).

## 1 Answer

Observe that $D^{-1/2}JD^{1/2}=D^{1/2}AD^{1/2},$ where $D^{1/2}=\mathrm{diag}(\sqrt{d_{i,i}}),$ so $J$ is similar to $D^{1/2}AD^{1/2},$ and they have the same eigenvalues. Also note that $D^{1/2}AD^{1/2}$ is Hermitian. By a theorem of Ostrowski, which can be found in Horn and Johnson's Matrix Analysis, if the eigenvalues are put into increasing order for all of the matrices, then $$\lambda_{k}(J)=\lambda_{k}(D^{1/2}AD^{1/2})=\theta_{k}\lambda_{k}(A),$$ where $\theta_{k}\in[\min_{i}d_{i,i},\max_{i}d_{i,i}].$