Interrelations between eigenvalues and diagonal entries of Hermitian matrices implying their block diagonality

Let $A=(a_{ij})^n_{i,j=1}$ be a Hermitian matrix and let $\lambda_1\geqslant\lambda_2\geqslant\dots\geqslant\lambda_n$ denote its eigenvalues. If $\sum^k_{i=1}=\lambda_i\sum^k_{i=1}a_{ii}$, $k<n$, then $A$ is known to be block diagonal. We show that this result easily follows from the Cauchy interlacing theorem, generalize it by introducing a convex strictly monotone function $f(t)$, and prove that in the positivedefinite case, the matrix diagonal entries can be replaced by the diagonal entries of a Schur complement. Bibliography: 4 titles.