Rank one matrices do not contribute to the failure of the finiteness property

TL;DR

This paper investigates conditions under which a set of matrices satisfies the finiteness property related to the joint spectral radius, highlighting the role of rank one matrices and providing exact formulas for specific matrix sets.

Contribution

It establishes sufficient conditions for the finiteness property based on rank one elements and generalizes existing results to sets with at most one non-rank-one matrix.

Findings

Sets lacking the finiteness property contain nonempty rank ≥ 2 matrices

The subset of rank ≥ 2 matrices shares the same joint spectral radius as the original set

Provides an exact formula for joint spectral radii when at most one matrix is not rank one

Abstract

The joint spectral radius of a bounded set of d times d real or complex matrices is defined to be the maximum exponential rate of growth of products of matrices drawn from that set. A set of matrices is said to satisfy the finiteness property if this maximum rate of growth occurs along a periodic infinite sequence. In this note we give some sufficient conditions for a finite set of matrices to satisfy the finiteness property in terms of its rank one elements. We show in particular that if a finite set of matrices does not satisfy the finiteness property, then the subset consisting of all matrices of rank at least two is nonempty, does not satisfy the finiteness property, and has the same joint spectral radius as the original set. We also obtain an exact formula for the joint spectral radii of sets of matrices which contain at most one element not of rank one, generalising a recent result of X. Dai.