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On pairs of spectrum maximizing products with distinct factor multiplicities

Victor Kozyakin

TL;DR

The paper addresses whether pairs of spectrum maximizing products with odd length can occur with different counts of the same-name factors. It develops a constructive framework based on $\tau$-permutable irreducible 2×2 matrix sets and extremal norms to produce such pairs, culminating in an analytic example where the max products have forms $BAA$ and $BBA$ (length 3). By constructing a polygonal unit ball $S$ and deriving precise parameter ranges for $\kappa$ and $\mu$, the authors prove the existence of a norm yielding $\bar{\rho}(\mathscr{A})=\lambda^{1/3}$ with the desired spectrum maximizing products. This approach provides an explicit alternative to previous numerically driven constructions and broadens the toolkit for generating non-unique spectrum maximizing products.

Abstract

Recently, Bochi and Laskawiec constructed an example of a set of matrices $\{A,B\}$ having two different (up to cyclic permutations of factors) spectrum maximizing products, $AABABB$ and $BBABAA$. In this paper, we identify a class of matrix sets for which the existence of at least one spectrum maximizing product with an odd number of factors automatically entails the existence of another spectrum maximizing product. Moreover, in addition to Bochi--Laskawiec's example, the number of factors of the same name (factors of the form $A$ or $B$) in these matrix products turns out to be different. The efficiency of the proposed approach is confirmed by constructing an example of a set of $2\times2$ matrices $\{A,B\}$ that has spectrum maximizing products of the form $BAA$ and $BBA$.

On pairs of spectrum maximizing products with distinct factor multiplicities

TL;DR

The paper addresses whether pairs of spectrum maximizing products with odd length can occur with different counts of the same-name factors. It develops a constructive framework based on -permutable irreducible 2×2 matrix sets and extremal norms to produce such pairs, culminating in an analytic example where the max products have forms and (length 3). By constructing a polygonal unit ball and deriving precise parameter ranges for and , the authors prove the existence of a norm yielding with the desired spectrum maximizing products. This approach provides an explicit alternative to previous numerically driven constructions and broadens the toolkit for generating non-unique spectrum maximizing products.

Abstract

Recently, Bochi and Laskawiec constructed an example of a set of matrices having two different (up to cyclic permutations of factors) spectrum maximizing products, and . In this paper, we identify a class of matrix sets for which the existence of at least one spectrum maximizing product with an odd number of factors automatically entails the existence of another spectrum maximizing product. Moreover, in addition to Bochi--Laskawiec's example, the number of factors of the same name (factors of the form or ) in these matrix products turns out to be different. The efficiency of the proposed approach is confirmed by constructing an example of a set of matrices that has spectrum maximizing products of the form and .
Paper Structure (11 sections, 5 theorems, 82 equations, 3 figures)

This paper contains 11 sections, 5 theorems, 82 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Basic example
  4. Extremal norm construction
  5. Selection of scaling parameter
  6. Proofs and auxiliary relations
  7. Proof of Theorem \ref{['thm:smp']}
  8. Location of the vertices of the polygon $S$
  9. Convexity of the polygon $S$
  10. Notes and comments
  11. Conclusion

Key Result

Lemma 2.2

An irreducible set of matrices $\mathscr{A}=\{A,B\}$ is permutable under some similarity transformation $\tau$ if and only if

Figures (3)

  • Figure 1: Polygon $S$ (solid line, gray background) and its images $\tilde{A} S$ (dashed line) and $\tilde{B} S$ (dash-dotted line)
  • Figure 2: Polygon $S$ (solid line, gray background) and its images $\tilde{A} S$ (dashed line) and $\tilde{B} S$ (dash-dotted line); $\varkappa=1.331$, $\mu=1.25$
  • Figure 3: Polygon $S$ from Example \ref{['ex:alt']} (solid line, gray background) and its images $\tilde{A} S$ (dashed line) and $\tilde{B} S$ (dash-dotted line); $\varkappa=1.331$, $\mu=1.07$

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Example 2.7
  • Remark 3.1
  • ...and 10 more