[Paper Review] Can an infinite left-product of nonnegative matrices be expressed in terms of infinite left-products of stochastic ones?

Alain Thomas|arXiv (Cornell University)|Aug 25, 2009

Matrix Theory and Algorithms8 references3 citations

TL;DR

This paper investigates whether infinite left-products of nonnegative matrices can be expressed via infinite left-products of stochastic matrices, even when the common right-eigenvector for eigenvalue 1 has zero entries. It establishes that convergence of the original product is equivalent to convergence of associated stochastic products on invariant subspaces, provided additional conditions on off-diagonal blocks are satisfied, extending known results beyond positive eigenvectors.

ABSTRACT

If a left-product $M_n... M_1$ of square complex matrices converges to a nonnull limit when $n o\infty$ and if the $M_n$ belong to a finite set, it is clear that there exists an integer $n_0$ such that the $M_n$, $n\ge n_0$, have a common right-eigenvector $V$ for the eigenvalue 1. Now suppose that the $M_n$ are nonnegative and that $V$ has positive entries. Denoting by $Δ$ the diagonal matrix whose diagonal entries are the entries of $V$, the stochastic matrices $S_n=Δ^{-1}M_nΔ$ satisfy $M_n... M_{n_0}=ΔS_n... S_{n_0}Δ^{-1}$, so the problem of the convergence of $M_n... M_1$ reduces to the one of $S_n... S_{n_0}$. In this paper we still suppose that the $M_n$ are nonnegative but we do not suppose that $V$ has positive entries. The first section details the case of the $2 imes2$ matrices, and the last gives a first approach in the case of $d imes d$ matrices.

Motivation & Objective

• To determine whether infinite left-products of nonnegative matrices can be characterized using infinite left-products of stochastic matrices, even when the common right-eigenvector for eigenvalue 1 has zero entries.
• To extend the classical reduction from nonnegative to stochastic matrices beyond the case of positive eigenvectors.
• To identify necessary and sufficient conditions for convergence of infinite left-products of nonnegative matrices using stochastic counterparts on invariant subspaces.
• To formalize the equivalence between convergence of the original matrix product and convergence of associated stochastic products on spectral subspaces, with additional conditions on off-diagonal blocks.

Proposed method

• For 2×2 matrices, the paper analyzes convergence by classifying cases based on the structure of the common right-eigenvector for eigenvalue 1, distinguishing between positive, zero, or mixed entries.
• It introduces a similarity transformation using diagonal matrices Δ derived from the eigenvector V, defining stochastic matrices Sₙ = Δ⁻¹MₙΔ to convert nonnegative matrices into stochastic ones.
• For d×d matrices, the method partitions the index set into blocks Kᵢ based on the support of each right-eigenvector Vᵢ, and constructs stochastic products Sₙ⁽ⁱ⁾ on each block Kᵢ.
• The convergence of the original product is shown to be equivalent to the convergence of the stochastic products Sₙ⁽ⁱ⁾⋯Sₙ₀⁽ⁱ⁾ on each block Kᵢ, plus convergence of off-diagonal block products involving Kᶜ.
• The method uses block matrix decomposition and product formulas to relate the original matrix product to products of triangular-by-blocks matrices, isolating the behavior on invariant subspaces.
• It employs the determinant and trace dynamics of stochastic products to analyze convergence, particularly focusing on whether |det Qₙ| tends to zero or not.

Experimental results

Research questions

• RQ1 Can the convergence of an infinite left-product of nonnegative matrices be reduced to the convergence of infinite left-products of stochastic matrices, even when the common right-eigenvector for eigenvalue 1 has zero entries?
• RQ2 What additional conditions are required to ensure that the convergence of the stochastic products on invariant subspaces implies convergence of the original nonnegative matrix product?
• RQ3 How does the structure of the right-eigenvectors (especially those with zero entries) affect the reducibility of nonnegative matrix products to stochastic ones?
• RQ4 Under what conditions on the off-diagonal blocks Mᴷᶜ, Mᴷ, and Mᴷ,ᴷᶜ does the convergence of the full product Mₙ⋯M₁ hold, given convergence on the invariant subspaces?

Key findings

• For 2×2 nonnegative matrices, convergence of the left-product Mₙ⋯M₁ is equivalent to one of three cases: (1) existence of a common positive right-eigenvector and convergence of the associated stochastic product, (2) common right-eigenvector (1,0)ᵀ with finite sum of bₙdₙ₋₁⋯dₙ₀ and convergent dₙ⋯dₙ₀, or (3) common right-eigenvector (0,1)ᵀ with finite sum of cₙaₙ₋₁⋯aₙ₀ and convergent aₙ⋯aₙ₀.
• In the d×d case, convergence of Mₙ⋯M₁ is equivalent to convergence of the stochastic products Sₙ⁽ⁱ⁾⋯Sₙ₀⁽ⁱ⁾ on each block Kᵢ corresponding to the support of a right-eigenvector Vᵢ, plus convergence of the off-diagonal block product ∑ᵢ₌ₙ₀ⁿ Mᴷⁿ⋯Mᴷⁱ⁺¹ Mᴷ,ᴷᶜⁱ Mᴷᶜⁱ⁻¹⋯Mᴷᶜⁿ₀.
• The condition on the sum ∑ᵢ₌ₙ₀ⁿ Mᴷⁿ⋯Mᴷⁱ⁺¹ Mᴷ,ᴷᶜⁱ Mᴷᶜⁱ⁻¹⋯Mᴷᶜⁿ₀ is necessary and cannot be omitted, as shown by a counterexample where this sum diverges despite convergence of the diagonal blocks.
• If the spectral radius of Mᴷᶜⁿ is less than 1 for all n, or if eigenvalues ≥1 vanish in products Mᴷᶜⁿ⁺ʰ⋯Mᴷᶜⁿ for fixed h, then the off-diagonal convergence condition is satisfied.
• The convergence of the normalized product Rₙ = Mₙ⋯M₁ / ||Mₙ⋯M₁|| implies that the limit matrix R has nonnull columns that are right-eigenvectors of each matrix M occurring infinitely often, with eigenvalue λ ≠ 0.
• The paper establishes that convergence of the normalized product Rₙ does not imply convergence of (1/λ)Mₙ⋯M₁, but the converse holds: if (1/λ)Mₙ⋯M₁ converges to a nonnull matrix, then Rₙ also converges.