Eigenvalue Bounds for Double Saddle-Point Systems

TL;DR

This paper derives bounds on the eigenvalues of double saddle-point matrices, considering various matrix properties and preconditioning effects, supported by numerical experiments.

Contribution

It provides new eigenvalue bounds for double saddle-point systems, including preconditioned matrices and approximations of Schur complements.

Findings

Eigenvalues are bounded within specific intervals away from zero.

Preconditioned matrices exhibit eigenvalue clustering.

Numerical experiments validate the theoretical bounds.

Abstract

We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.