Solution
= Solution
$Pu=0$, hence $HPu=0$. In the diagonal <basis>, $f(\lambda)=1-\sum_j u_j^2\mu_j/(\mu_j-\lambda)=-\lambda\sum_j u_j^2/(\mu_j-\lambda)$. Between consecutive $\mu_j$ its nonzero factor is strictly monotone from one infinite sign to the other, giving one root; zero weights give equality at endpoints. Hence $\mu_1\le\sigma_1\le\mu_2\le\cdots\le\sigma_{n-1}\le\mu_n$.