On the eigenvalues and diagonal entries of a Hermitian matrix (Q1208295)

Let \(A=[a_{ij}]\) be an \(n\times n\) Hermitian matrix, and let \(\lambda_ 1\geq\lambda_ 2 \geq\dots\geq \lambda_ n\) be its eigenvalues. Suppose that, for some \(k\), \(\lambda_ 1+\lambda_ 2 +\dots+\lambda_ k=a_{11}+ a_{22}+\dots+ a_{kk}\). Then the authors show that \(A\) is block diagonal of the form \(\text{diag}(A_ 1,A_ 2)\) where the blocks \(A_ 1\), \(A_ 2\) are \(k\times k\) and \((n-k)\times(n- k)\), respectively.