On the Condition of Symmetric Quasi-Definite Matrices

A promising approach to solving sparse systems of equations involves creating an augmented system that is symmetric quasi-definite. The attraction of this approach stems from being able to use powerful algorithms and efficient data structures that have been developed for sparse positive definite systems. In this article, bounds are derived for the condition number of the Schur complements that emerge when symmetric Gaussian elimination is applied to these quasi-definite matrices. An application of these bounds to solving regularized linear least squares problems is discussed. For the corresponding linear systems, the bounds may be used to justify a reordering based solely on sparsity preservation considerations.