Rank-one Characterization of Joint Spectral Radius of Finite Matrix Family

TL;DR

This paper investigates the joint spectral radius of finite matrix sets using rank-one approximations, deriving formulas and new characterizations, with applications and numerical examples demonstrating the approach.

Contribution

It introduces a novel rank-one approximation method for analyzing the joint spectral radius of matrix families, including formulas and characterizations for general cases.

Findings

Finite set matrices with at most one higher-rank element satisfy the finiteness property.

Derived formulas for spectral radius computation in specific matrix classes.

Numerical examples demonstrate the effectiveness of the proposed approach.

Abstract

In this paper we study the joint/generalized spectral radius of a finite set of matrices in terms of its rank-one approximation by singular value decomposition. In the first part of the paper, we show that any finite set of matrices with at most one element's rank being greater than one satisfies the finiteness property under the framework of (invariant) extremal norm. Formula for the computation of joint/generalized spectral radius for this class of matrix family is derived. Based on that, in the second part, we further study the joint/generalized spectral radius of finite sets of general matrices through constructing rank-one approximations in terms of singular value decomposition, and some new characterizations of joint/generalized spectral radius are obtained. Several benchmark examples from applications as well as corresponding numerical computations are provided to illustrate the approach.