## The Schur complement of strictly doubly diagonally dominant matrices and its application.(English)Zbl 1248.15018

The Schur complements of doubly diagonally dominant ($$DD_n$$) matrices are doubly diagonally dominant, a result of B. Li and M. J. Tsatsomeros [Linear Algebra Appl. 261, 221–235 (1997; Zbl 0886.15027)]. The authors obtain an estimate for the doubly diagonally dominant degree on the Schur complement of strictly doubly diagonally dominant matrices ($$SDD_n$$). This extends the result of Li and Tsatsomeros. As an application they show that the eigenvalues of the Schur complements are located in the Brauer ovals of Cassini of the original matrices under certain conditions. They also obtain an upper bound for the infinity norm on the inverse on the Schur complement of $$SDD_n$$. They then give an iteration called the Schur-based iteration which can solve large scale linear systems through reducing the order by the Schur complement. Comparing some methods in the literature, their computation is faster in reducing the order of large matrices.

### MSC:

 15A42 Inequalities involving eigenvalues and eigenvectors 15A45 Miscellaneous inequalities involving matrices 15B48 Positive matrices and their generalizations; cones of matrices 65F10 Iterative numerical methods for linear systems 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory

Zbl 0886.15027
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