The Schur complement of strictly doubly diagonally dominant matrices and its application

It is known that the Schur complements of doubly diagonally dominant matrices are doubly diagonally dominant. In this paper, we obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices. Then, as an application we obtain that the eigenvalues of the Schur complements are located in the Brauer Ovals of Cassini of the original matrices under certain conditions. As another application, we obtain an upper bound for the infinity norm on the inverse on the Schur complement of strictly doubly diagonally dominant matrices. Further, based on the derived results, we give a kind of iteration called the Schur-based iteration, which can solve large scale linear systems though reducing the order by the Schur complement and can compute out the results faster.