Quasi-Perron-Frobenius property of a class of saddle point matrices

TL;DR

This paper introduces the quasi-Perron-Frobenius property for saddle point matrices, showing under certain conditions that their spectral radius equals the maximum eigenvalue, with numerical tests confirming the theory.

Contribution

It establishes a new spectral property for saddle point matrices, extending Perron-Frobenius concepts to a broader class of matrices in scientific computing.

Findings

Spectral radius equals maximum eigenvalue when C is zero.

Numerical tests confirm theoretical predictions.

Condition for quasi-Perron-Frobenius property is sufficient, not necessary.

Abstract

The saddle point matrices arising from many scientific computing fields have block structure $ W= \left(\begin{array}{cc} A & B\\ B^T & C \end{array} \right) $, where the sub-block $A$ is symmetric and positive definite, and $C$ is symmetric and semi-nonnegative definite. In this article we report a unobtrusive but potentially theoretically valuable conclusion that under some conditions, especially when $C$ is a zero matrix, the spectral radius of $W$ must be the maximum eigenvalue of $W$. This characterization approximates to the famous Perron-Frobenius property, and is called quasi-Perron-Frobenius property in this paper. In numerical tests we observe the saddle point matrices derived from some mixed finite element methods for computing the stationary Stokes equation. The numerical results confirm the theoretical analysis, and also indicate that the assumed condition to make the saddle point matrices possess quasi-Perron-Frobenius property is only sufficient rather than necessary.