Inequalities connecting the eigenvalues of a hermitian matrix with the eigenvalues of complementary principal submatrices

Abstract

Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α 1 ≥ … ≥ α a , β 1 ≥ … ≥ β b , γ 1 ≥ … ≥ γ n , respectively. In this paper four classes of inequalities are proved comparing the α i . and β j with the γ k . The simplest of these is: if the subscripts i s , j s satisfy 1 ≤ i 1 < … < i m ≤ α, 1 ≤ j 1 < … < j m ≤ b.