Submitted to the proceedings of the ninth international conference on domain decomposition methods (Bergen-Norway-1996).

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddlepoint problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen ’00], [de Sturler and Liesen ’03] and [Siefert and de Sturler ’04]. This approach can be applied in similar preconditioners as well. W...

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddlepoint problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen ’00], [de Sturler and Liesen ’03] and [Siefert and de Sturler ’04]. This approach can be applied in similar preconditioners as well. W...

Let $(A)$ be a complex $(ntimes n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(alpha)$ denoted by $(S_{alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{alpha})$.The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruen...

The purpose of this paper is to revisit Hua’s matrix equality (and inequality) through the Schur complement. We present Hua’s original proof and two new proofs with some extensions of Hua’s matrix equality and inequalities. The new proofs use a result concerning Schur complements and a generalization of Sylvester’s law of inertia, each of which is useful in its own right.

This paper proposes the definition of the generalized Schur complement on nonstrictly diagonally dominant matrices and general H−matrices by using a particular generalized inverse, and then, establishes some significant results on heredity, nonsingularity and the eigenvalue distribution for these generalized Schur complements.

We consider the Geršgorin disc separation from the origin for (doubly) diagonally dominant matrices and their Schur complements, showing that the separation of the Schur complement of a (doubly) diagonally dominant matrix is greater than that of the original grand matrix. As application we discuss the localization of eigenvalues and present some upper and lower bounds for the determinant of dia...

This paper proposes the definition of the generalized Schur complement on nonstrictly diagonally dominant matrices and general H−matrices by using a particular generalized inverse, and then, establishes some significant results on heredity, nonsingularity and the eigenvalue distribution for these generalized Schur complements.