The Schur Complement of γ-Dominant Matrices

In this paper, we present the γ (product γ )-diagonally dominant degree on the Schur complement of γ (product γ )-diagonally dominant matrices. Consequently, the disc theorems for the Schur complement of γ (product γ )-diagonally dominant matrices are obtained using the diagonally dominant degree on Schur complements, which improves and extends some related results. Further, to solve large-scale linear systems, we give the spectral radius estimates for the inverse matrices of γ (product γ )-diagonally dominant matrices and their Schur complements. When the coefficient matrix of linear system is γ (product γ )-diagonally dominant matrix, according to the special properties and the structures of γ (product γ )-diagonally dominant matrices, we design a class of special iteration for solving equations. Combining the spectral iteration with the Schur-based iterative method, we obtain a new iterative method called the Schur-based inverse algorithm, which reduces the order of the matrix and has a good convergence. The numerical examples are given to verify the validity of our results.