Solutions of permanental equations regarding stochastic matrices
type: Item
instance of: scholarly article
title: Solutions of permanental equations regarding stochastic matrices
zbMATH Open document ID: 0959.15028
DOI: 10.1016/S0898-1221(00)00170-X
author: B. Gyires
publication date: 2000-11-13T00:00:00Z
review text: Let \(A\) be an \(n\times n\) doubly stochastic matrix, \(1\leq k\leq n\). For \(1\leq i_1< i_2<\dots <i_k\leq n\) and \(1\leq j_1<j_2<\dots <j_k\leq n\), let \(A_{i_1\dots i_k}^{j_1\dots j_k}\) be the \(k\times k\) matrix consisting of the entries on the \(i_1\)th, \(i_2\)th, \(\dots\), \(i_k\)th rows and the \(j_1\)th, \(j_2\)th, \(\dots\), \(j_k\)th columns of \(A\). Define \(t_0(A)=1\), and \[ t_k(A)=\frac {\sum_{i_1<\dots <i_k, j_1<\dots <j_k} \text{Permanent}(A_{i_1\dots i_k}^{j_1\dots j_k})}{\binom nk^2}. \] The paper shows, by means of the above operators and concominants, that if the upper and lower permanents of the product of two \(n\times n\) doubly stochastic matrices are equal then at least one factor is the matrix with all entries equal to \(\frac 1n\). In addition, a few results about the means of permanents of doubly stochastic matrices are obtained.
reviewed by: Huang Wenxue
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 1528537
zbMATH Keywords: concominant
zbMATH Keywords: doubly-stochastic matrix
zbMATH Keywords: upper and lower permanents
zbMATH Keywords: permanent equations
zbMATH Keywords: means of permanents
zbMATH Keywords: van der Waerden's conjecture
MaRDI profile type: Publication
cites work: Permanents
cites work: Elementary proof for a Van der Waerden's conjecture and related theorems
cites work: The solution of van der Waerden's problem for permanents
cites work: Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix
cites work: Q4184985
cites work: Q3900194
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 1528537