An explicit formula for singular values of the Sylvester-Kac matrix
type: Item
instance of: scholarly article
title: An explicit formula for singular values of the Sylvester-Kac matrix
zbMATH Open document ID: 1121.15012
author: Tibor Boros
author: Pál Rózsa
published in: Linear Algebra and its Applications
publication date: 2007-03-09T00:00:00Z
review text: Let \(P(n)=(p_{ij}(n))\) be the \((n+1)\times (n+1)\) tridiagonal centrosymmetric matrix satisfying \(p_{i+1,i}(n)=i\) and \(p_{i,i}(n)=0\), for all \(i\). The authors show that, for all nonnegative integers \(m\), (i) the singular values of \(P(2m)\) include the numbers \(\sqrt{(2m+1)^2-(2i+1)^2}\), \(i=0,1,2,\dots ,m\), and (ii) the singular values of \(P(2m+1)\) all have multiplicity two. The authors first use the Perron-Frobenius canonical form to reduce the singular value problem for \(P(n)\) to the singular value problems for two smaller matrices, a step similar to the usual splitting technique for centrosymmetric matrices. They then find the singular values of one of these smaller matrices by finding all nontrivial polynomial solutions of a related differential equation. These polynomials provide the elements of the singular vectors. The paper also includes a brief discussion of the work of J. J. Sylvester, M. Kac and others on the eigenvalue problem for \(P(n)\).
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 5132607
zbMATH Keywords: stochastic matrices
zbMATH Keywords: centrosymmetric matrices
zbMATH Keywords: singular values
zbMATH Keywords: Frobenius series
zbMATH Keywords: Perron-Frobenius canonical form
zbMATH Keywords: polynomial solutions
zbMATH Keywords: differential equation
zbMATH Keywords: singular vectors
zbMATH Keywords: eigenvalue
reviewed by: Alan L. Andrew
MaRDI profile type: Publication
full work available at URL: j.laa.2006.10.008
OpenAlex ID: W1977936983
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DOI: 10.1016/J.LAA.2006.10.008
description: scientific article; zbMATH DE number 5132607