On a conjecture of E. Dittert
type: Item
instance of: scholarly article
title: On a conjecture of E. Dittert
zbMATH Open document ID: 0628.15018
DOI: 10.1016/0024-3795(87)90032-2
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \(K_ n\) denote the set of \(n\times n\) nonnegative matrices whose entries have sum n. For an \(n\times n\) matrix \(X=(x_{ij})\in K_ n\), put \[ \phi (x)=\prod^{n}_{i=1}\sum^{n}_{j=1}x_{ij}+\prod^{n}_{j=1}\sum\;sp{n}_{i=1}x_{ij}-per X \] and for each i, j, put \(\phi_{ij}(x)=\prod_{k\neq i}r_ k+\prod_{k\neq j}c_ k-per X(i| j).\) Here \(r_ k\) is the kth row sum and \(c_ k\) is the kth column sum of X and X(i\(| j)\) is the (n-1)\(\times (n-1)\) matrix obtained by deleting the ith row and jth column of X, \(n>1\). If \(n=1\), \(X=(1)\) and \(\phi (X)=\phi_{11}(X)=1.\) A matrix \(A\in K_ n\) is \(\phi\)-maximizing if \(\phi (A)\geq \phi (X)\) for all \(X\in K_ n\). E. Dittert conjectured that if \(A\in K_ n\) is \(\phi\)-maximizing then \(A=J_ n\), the \(n\times n\) matrix whose entries all equal \(n^{-1}\). The author proves that the conjecture is true for \(n=3\) and shows that for any n if \(A\in K_ n\) is \(\phi\)-maximizing, then \(\phi_{ij}(A)=\phi (A)\) whenever \(\alpha_{ij}>0\) and \(\phi_{ij}(A)\leq \phi (A)\) whenever \(\alpha_{ij}=0\).
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 4021362
zbMATH Keywords: phi-maximizing matrix
zbMATH Keywords: permanent function
zbMATH Keywords: nonnegative matrices
author: Suk-Geun Hwang
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90032-2
OpenAlex ID: W2048566573
Wikidata QID: Q123167037
cites work: A note on a conjecture on permanents
cites work: The monotonicity of and the Đoković conjectures on permanents of doubly stochastic matrices
cites work: Theory of permanents 1978–1981
cites work: A problem related to the van der waerden permanent theorem*
description: scientific article; zbMATH DE number 4021362