[Paper Review] Eigenvalue problem for some special class of anti-triangular matrices

Hiroyuki Ochiai, Makiko Sasada+2 more|arXiv (Cornell University)|Mar 14, 2014

Matrix Theory and Algorithms8 references4 citations

TL;DR

This paper introduces a special class of anti-triangular matrices for which eigenvalues are explicitly determined by their diagonal elements, with the eigenvalues given as alternating signs of the diagonal entries. The key contribution is a characterization of matrices with this 'anti-diagonal eigenvalue property' using probability measures, particularly linking them to beta, negative binomial, and Poisson distributions via a generating function framework.

ABSTRACT

We study the eigenvalue problem for some special class of anti-triangular matrices. Though the eigenvalue problem is quite classical, as far as we know, almost nothing is known about properties of eigenvalues for anti-triangular matrices. In this paper, we show that there is a nice class of anti-triangular matrices whose eigenvalues are given explicitly by their elements. Moreover, this class contains several interesting subclasses which we characterize in terms of probability measures. We also discuss the application of our main theorem to the study of interacting particle systems, which are stochastic processes studied in extensive literature.

Motivation & Objective

• To address the lack of known eigenvalue properties for anti-triangular matrices, a class for which eigenvalues can be explicitly computed.
• To characterize a class of anti-triangular matrices whose eigenvalues are given by alternating signs of their diagonal entries.
• To establish connections between such matrices and probability measures, especially in the context of exchangeable sequences and Pólya urn models.
• To apply the main results to interacting particle systems, particularly in estimating convergence rates via eigenvalue analysis.

Proposed method

• Introduces the concept of 'anti-diagonal eigenvalue property' for matrices X such that XG is similar to a diagonal matrix with entries (−1)^i x_ii.
• Uses the matrix G, which satisfies G^2 = I, to transform lower triangular matrices into anti-triangular form.
• Defines a generating function g(i) = x_ii / x_{i-1,i-1} and derives recurrence relations for sequences V_i(z) and Ṽ_i(z) to analyze the eigenvalue condition.
• Applies the identity theorem for rational functions to extend functional identities from real intervals to the complex plane.
• Characterizes the class of functions g for which the matrix A_g belongs to the space V_P(∞), showing that only specific forms (G_t or identity) satisfy the condition.
• Links the resulting classes to known probability distributions: beta, negative binomial, and Poisson, via the generating function g.

Experimental results

Research questions

• RQ1 For which class of anti-triangular matrices can eigenvalues be explicitly expressed in terms of their diagonal elements?
• RQ2 What conditions on the generating function g(i) ensure that the associated matrix A_g has the anti-diagonal eigenvalue property?
• RQ3 How are these matrices related to classical probability models such as the Pólya urn and exchangeable sequences?
• RQ4 What is the precise characterization of probability measures ν for which the matrix R̃_ν belongs to the space V_P(∞)?
• RQ5 Can the eigenvalue structure of anti-triangular matrices be used to analyze convergence rates in interacting particle systems?

Key findings

• The eigenvalues of the matrix TG, where T is a lower triangular matrix with t_ij = 1/(i+1) for i ≥ j, are explicitly given as (1, -1/2, 1/3, -1/4, ..., (-1)^n/(n+1)).
• Similarly, for the matrix SG with s_ij = binom(i,j)/2^i for i ≥ j, the eigenvalues are (1, -1/2, 1/4, -1/8, ..., (-1)^n/2^n).
• A matrix X has the anti-diagonal eigenvalue property if and only if its generating function g(i) satisfies g(i) = α G_t(i) or g(i) = α i for some t > 0 and α > 0.
• The class of matrices with this property corresponds exactly to the set of matrices derived from beta distributions with parameters (t,t), including the limit case t = ∞.
• The matrix R̃_ν belongs to V_P(∞) if and only if ν is a negative binomial or Poisson distribution, with explicit parameterization in terms of g_ν.
• For all such cases, the key identity (3.3) holds, ensuring the validity of the eigenvalue characterization.