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

Let A=(a_ij) _i,j ⁿ =1 be a Hermitian matrix and let denote its eigenvalues. If , 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. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 153–158. Translated by L. Yu. Kolotilina.