A note on the variation of permanents
type: Item
instance of: scholarly article
title: A note on the variation of permanents
zbMATH Open document ID: 0653.15005
DOI: 10.1016/0024-3795(88)90196-6
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://pub.uni-bielefeld.de/record/1780558
review text: \textit{R. Bhatia} [ibid. 62, 269-276 (1984; Zbl 0552.15014)] showed that for any two \(n\times n\)-matrices A, B the inequality \((*)\quad | per(A)-per(B)| \leq n\| A-B\|_ 2\max (\| A\|_ 2,\| B\|_ 2)^{n-1}\) holds. Using quite elementary methods, the author of the paper under review proves the analogous result for \(\|.\|_ 2\) being replaced by the row-sum and the column-sum norms \(\| A\|_{\infty}=\max (| a_{i1}|,...,| a_{in}|),\) respectively \(\| A\|_ 1=\max_{k}(| a_{1k}|,...,| a_{nk}|).\) He conjectures that (*) is still true if \(\|.\|_ 2\) is replaced by any operator norm for \(n\times n\)-matrices. In this case, a determinantal analogue has been settled by \textit{S. Friedland} [Linear Multilinear Algebra 12, 81-98 (1982; Zbl 0491.15002)].
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 26B30
zbMATH DE Number: 4065173
zbMATH Keywords: permanent of a matrix
zbMATH Keywords: variation
zbMATH Keywords: operator norm
author: Ludwig Elsner
MaRDI profile type: Publication
OpenAlex ID: W1971984572
cites work: Variation of symmetric tensor powers and permanents
cites work: Variation of tensor powers and spectrat
cites work: Permanents
description: scientific article; zbMATH DE number 4065173