The eigenvalue distribution on Schur complement of nonstrictly diagonally dominant matrices and general H-matrices

Abstract

The paper studies the eigenvalue distribution of Schur complements of some special
matrices, including nonstrictly diagonally dominant matrices and general H−matrices. Zhang, Xu, and Li [Theorem 4.1, The eigenvalue distribution on Schur complements of H-matrices. Linear Algebra Appl., 422:250–264, 2007] gave a condition for an n×n diagonally dominant matrix A to have |J_R+(A)| eigenvalues with positive real part and |J_{R_} (A)| eigenvalues with negative real part, where |J_R+ (A)| (|J_{R_} (A)|) denotes the number of diagonal entries of A with positive (negative) real part. This condition is applied to establish some results about the eigenvalue distribution for the Schur complements of nonstrictly diagonally dominant matrices and general H−matrices with complexdiagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, · · · , n} arepresented so that the Schur complement A/α of A has |J_R+ (A)|−|J^α_R+ (A)| eigenvalues with positivereal part and |J_{R_} (A)| − |J^α_{R_} (A)| eigenvalues with negative real part, where |J^α_R+ (A)| (|J^α_{R_} (A)|) denotes the number of diagonal entries of the principal submatrix A(α) of A with positive (negative) real part.