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Joint spectral radius and forbidden products

Alexander Vladimirov

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.

Joint spectral radius and forbidden products

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 (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 of real 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 of matrices, even complex. For pairs of complex matrices we have numerical evidence that 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.

Paper Structure

This paper contains 12 sections, 3 theorems, 26 equations, 1 figure.

Table of Contents

  1. Joint spectral radius
  2. Binary words
  3. List of words
  4. Frequency lists
  5. Forbidden words
  6. Spectral radius as a function of trace
  7. Reduced problems
  8. Fricke polynomials
  9. Inequalities
  10. General values of determinants
  11. Forbidden patterns
  12. Double rotations

Key Result

Theorem 5.1

For any pair of real $2\hbox{$\times$} 2$ matrices $\{A,B\}$, the product never delivers a single maximum (up to cyclic shifts and mirror images) to the normalized spectral radius among all the products of length $8$ or less.

Figures (1)

  • Figure 1: Areas of maximality

Theorems & Definitions (5)

  • Theorem 5.1
  • Lemma 6.1
  • proof
  • Lemma 9.1
  • Definition 11.1