The finite-step realizability of the joint spectral radius of a pair of $d\times d$ matrices one of which being rank-one

TL;DR

This paper proves that for a pair of real matrices where one is rank-one, the joint spectral radius can be realized in finite steps, making stability decision-making algorithmically feasible.

Contribution

It establishes the finite-step realizability of the joint spectral radius for matrix pairs with one rank-one matrix, enabling algorithmic stability analysis.

Findings

Existence of a finite-length word with spectral finiteness property

Stability is algorithmically decidable for the considered matrix pairs

Provides a constructive approach to spectral radius computation

Abstract

We study the finite-step realizability of the joint/generalized spectral radius of a pair of real $d\times d$ matrices, one of which has rank 1. Then we prove that there always exists a finite-length word for which there holds the spectral finiteness property for the set of matrices under consideration. This implies that stability is algorithmically decidable in our case.