Joint spectral radius and forbidden products

TL;DR

This paper investigates forbidden products related to the joint spectral radius of matrix pairs, proving certain products are never maximal for real 2x2 matrices and providing numerical evidence for complex matrices.

Contribution

It proves that a specific product is forbidden for real 2x2 matrices and explores spectral properties of isospectral products, extending understanding of joint spectral radius behavior.

Findings

The product AABABABB is forbidden for real 2x2 matrices.

Circular shifts and mirror images of the product share the same spectral radius.

Numerical evidence suggests the product is also forbidden for complex matrices.

Abstract

We address the problem of finite products that attain the joint spectral radius of a finite number of square matrices. Up to date the problem of existence of "forbidden products" remained open. We prove that the product $AABABABB$ (together with its circular shifts and their mirror images) never delivers the strict maximum to the joint spectral radius if we restrict consideration to pairs $\{A,B\}$ of real $2\by 2$ matrices. Under this restriction circular shifts and their mirror images constitute the class of isospectral products and hence they all have the same spectral radius for any pair $\{A,B\}$ of $2\by 2$ matrices, even complex. For pairs of complex matrices we have numerical evidence that $AABABABB$ is still a fobidden product. A couple of binary words that encode products from this isospectral class also happen to be the shortest forbidden patterns in the parametric family of double rotations.