# Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets - Mathematics > Rings and Algebras

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Abstract: The problem of construction of Barabanov norms for analysis of properties ofthe joint generalized spectral radius of matrix sets has been discussed in anumber of publications. The method of Barabanov norms was the key instrument indisproving the Lagarias-Wang Finiteness Conjecture. The related constructionswere essentially based on the study of the geometrical properties of the unitballs of some specific Barabanov norms. In this context the situation when onefails to find among current publications any detailed analysis of thegeometrical properties of the unit balls of Barabanov norms looks a bitparadoxical. Partially this is explained by the fact that Barabanov norms aredefined nonconstructively, by an implicit procedure. So, even in simplest casesit is very difficult to visualize the shape of their unit balls. The presentwork may be treated as the first step to make up this deficiency. In the papertwo iteration procedure are considered that allow to build numericallyBarabanov norms for the irreducible matrix sets and simultaneously to computethe joint spectral radius of these sets.

Author: ** Victor Kozyakin**

Source: https://arxiv.org/