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The finiteness conjecture for $3 \times 3$ binary matrices

Thomas Mejstrik

TL;DR

This paper extends the invariant polytope algorithm by integrating mixed numeric/symbolic computations and additive limit-matrix input to broaden its applicability and termination speed. It leverages these enhancements to automatically prove the finiteness property for all pairs of binary $3\times3$ matrices and sign $2\times2$ matrices, addressing a longstanding question in JSR theory. The approach combines spectral-maximizing-product identification with invariant polytopes and uses symmetry reductions to manage combinatorial complexity, enabling practical verification of otherwise intractable cases. The work provides a MATLAB implementation (tjsr) and demonstrates the method’s effectiveness with concrete examples and extensive computational results.

Abstract

The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many applications in problems of functional analysis, approximation theory, combinatorics, etc. In this paper we propose a modification of the invariant polytope algorithm enlarging the class of problems to which it is applicable. Precisely, we introduce mixed numeric and symbolic computations. A further minor modification of augmenting the input set with additional matrices speeds up the algorithm in certain cases. With this modifications we are able to automatically prove the finiteness conjecture for all pairs of binary $3\times 3$ matrices and sign $2\times 2$ matrices.

The finiteness conjecture for $3 \times 3$ binary matrices

TL;DR

This paper extends the invariant polytope algorithm by integrating mixed numeric/symbolic computations and additive limit-matrix input to broaden its applicability and termination speed. It leverages these enhancements to automatically prove the finiteness property for all pairs of binary matrices and sign matrices, addressing a longstanding question in JSR theory. The approach combines spectral-maximizing-product identification with invariant polytopes and uses symmetry reductions to manage combinatorial complexity, enabling practical verification of otherwise intractable cases. The work provides a MATLAB implementation (tjsr) and demonstrates the method’s effectiveness with concrete examples and extensive computational results.

Abstract

The invariant polytope algorithm was a breakthrough in the joint spectral radius computation, allowing to find the exact value of the joint spectral radius for most matrix families~\cite{GP2013,GP2016}. This algorithm found many applications in problems of functional analysis, approximation theory, combinatorics, etc. In this paper we propose a modification of the invariant polytope algorithm enlarging the class of problems to which it is applicable. Precisely, we introduce mixed numeric and symbolic computations. A further minor modification of augmenting the input set with additional matrices speeds up the algorithm in certain cases. With this modifications we are able to automatically prove the finiteness conjecture for all pairs of binary matrices and sign matrices.
Paper Structure (13 sections, 10 theorems, 7 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 10 theorems, 7 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Overview and main results
  3. Notation
  4. The invariant polytope algorithm
  5. Mixed numeric/symbolic computations
  6. Limit matrices
  7. The finiteness conjecture
  8. Diminishing the number of cases
  9. Implementation notes
  10. List of cases
  11. 2 times 2 binary matrices
  12. 2 times 2 sign matrices
  13. 3 times 3 binary matrices

Key Result

Theorem 2.1

Let $\mathcal{A}=\left\lbrace A_j\in\mathbb{R}^{s\times s}:j=1,\ldots,J \right\rbrace$ be a finite set of matrices. The ipa terminates if and only if the set $\mathcal{A}$

Figures (3)

  • Figure 1: Various convex hulls. $v_1=12$, $v_2=21$, $v_3=0.50.5$, $v_4=2i$, $v_5=i2$.
  • Figure 2: Tree generated by the ipa with mixed numeric/symbolic computations for Example \ref{['ex_prod_unitmatrix']}. The starting vector $v_0$ is the leading eigenvector of $A_2$. Arrows depict how vertices are mapped under the given matrix product. Vertices plotted as $\bullet$ (instead written as text), are mapped to the interior of the polytope $P=\operatorname{co}_{\operatorname{s}}\left\lbrace v_0, A_1 v_0, A_1 A_1 v_0, A_2 A_1 v_0, A_2 A_2 A_1 v_0 \right\rbrace$.
  • Figure 3: Invariant polytope for the matrices $A_1 = 0101,\quad A_2 = 1\space01-1.$ of Example \ref{['ex_tjsr']}. The cross $\times$ denotes the origin, the dots $\bullet$ the leading eigenvectors of the matrices (and s.m.p.s) $A_1$ and $A_2$.

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 2.1
  • Theorem 2.1: GP2016
  • Example 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3.1
  • ...and 20 more