Positive eigenvalues and two-letter generalized words
type: Item
instance of: scholarly article
title: Positive eigenvalues and two-letter generalized words
zbMATH Open document ID: 0997.15009
DOI: 10.13001/1081-3810.1069
author: Christopher J. Hillar
author: Charles R. Johnson
author: Ilya M. Spitkovsky
publication date: 2002-03-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0504573
full work available at URL: https://eudml.org/doc/122134
full work available at URL: 9.html
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 03D40
Mathematics Subject Classification ID: 06B25
Mathematics Subject Classification ID: 08A50
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 68R15
zbMATH DE Number: 1716192
zbMATH Keywords: positive definite matrices
zbMATH Keywords: positive eigenvalues
zbMATH Keywords: generalized word
zbMATH Keywords: letter
zbMATH Keywords: projections
MaRDI profile type: Publication
OpenAlex ID: W2139581260
arXiv ID: math/0504573
published in: ELA. The Electronic Journal of Linear Algebra
review text: A so-called generalized word of class \(N\) in two letters \(A\) and \(B\) is defined as an expression \(W(A,B)=A^{\alpha_1}B^{\beta_1}A^{\alpha_2} B^{\beta_2}\dots A^{\alpha_N}B^{\beta_N}\), where the exponents \(\alpha_1, \beta_1,\dots,\alpha_N,\beta_N\) are nonzero real numbers. Here, the eigenvalues of words \(W(A,B)\) are studied when \(A\) and \(B\) are replaced by positive definite \(n \times n\) matrices. It is of special interest to know when words \(W(A,B)\) have positive eigenvalues. NEWLINENEWLINENEWLINEIn the previously studied case when \(N=1\) and all exponents were positive the eigenvalues were positive. In this work this case is extended on the situation when the exponents have mixed signs. In several theorems conditions on \(A\), \(B\) (e. g. multiplicity of eigenvalues) and on a sign pattern of exponent sequences for the cases \(n=2,3\) and arbitrary \(n\) for \(W(A,B)\) to have positive eigenvalues are found.
description: scientific article; zbMATH DE number 1716192