Multiplicities of the eigenvalues of a perturbed matrix Suppose $S$ is a symmetric and positive definite matrix in $\mathcal{R}^{n\times n}$. Suppose that $D$ is a diagonal matrix with diagonal $[1,a^{-1},\cdots,a^{-n+1}]$ for some positive real $a$. If S has $n$ different eigenvalues, then is it true that $DS$ has $n$ different eigenvalues? What can be said about the eigenvalues of $DS$ or their multiplicities?

No. Consider, e.g. the case $0
I don't know what can be said about the multiplicities of the eigenvalues of $DS$ in general, but we do know that all eigenvalues of $DS$ are positive, as $DS$ is similar to $D^{1/2}SD^{1/2}$, which is positive definite.