Classical Results and Problems

Abstract

In this chapter we review major results on the joint spectral radius. Our goal is to remain concise, but at the same time exhaustive and self-contained. We begin by analyzing in detail the growth of matrix products, and by presenting the concept of extremal norms. Existence of extremal norms is an encouraging result, since it is easier to evaluate the joint spectral radius when an extremal norm is available. We found it natural to follow with calculability/complexity theorems, which are on the other hand discouraging. In a subsequent section, we present methods of computation and approximation of the joint spectral radius. In view of the negative results of the second section, the reader shall not be surprised to find algorithms whose efficiency is often rather poor (at least theoretically). In the last section of this chapter we present a fascinating question: the finiteness property.