Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices
type: Item
instance of: scholarly article
title: Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices
zbMATH Open document ID: 0726.15016
DOI: 10.1016/0024-3795(90)90266-F
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: A polygon in the complex plane is called cyclic if it is the convex hull of 1, \(\lambda\), \(\lambda^ 2,...\), for some complex number \(\lambda\). The regions where \(\lambda\) should lie in order that this polygon be n- sided are determined. As an application of this result the region in the complex plane in which the roots of the equation \(\lambda^{n-1}+a_ 1\lambda^{n-2}+...+a_{n-1}=0\) lie when the coefficients satisfy the inequalities \(1\geq a_ 1\geq a_ 2\geq...\geq a_{n-1}\geq 0\) are determined. As another application, it is shown that a conjecture of \textit{N. Dmitriev} and \textit{E. Dynkin} [Izv. Akad. Nauk SSSR, Ser. Mat. 10, 167- 184 (1946)] concerning eigenvalues of stochastic matrices is false for 6 by 6 matrices, and the statement of a theorem by \textit{F. I. Karpelevich} [Izv. Akad. Nauk SSSR, Ser. Mat. 15, 361-383 (1951; Zbl 0043.016)] describing the region where the eigenvalues on n by n stochastic matrices lie is simplified.
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 52A10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 4198211
zbMATH Keywords: cyclic polygon
zbMATH Keywords: conjecture of Dmitriev and Dynkin
zbMATH Keywords: Karpelevich theorem
zbMATH Keywords: eigenvalues of stochastic matrices
author: Dragomir Ž. Đoković
MaRDI profile type: Publication
cites work: Q4745933
cites work: Q5834413
cites work: Q3259107
cites work: Q5805016
cites work: Q4039989
cites work: Q5614013
full work available at URL: 0024-3795(90)90266-f
OpenAlex ID: W2103023262
description: scientific article; zbMATH DE number 4198211