The eigenvalue distribution on Schur complements of H-matrices

The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273–275] gives conditions for an n × n matrix A ∈ SDn ∪ IDn to have | J R + ( A ) | eigenvalues with positive real part, and | J R - ( A ) | eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has | J R + ( A ) | - | J R + α ( A ) | eigenvalues with positive real part and | J R - ( A ) | - | J R - α ( A ) | eigenvalues with negative real part.