Preservers of matrix pairs with a fixed inner product value
type: Item
instance of: scholarly article
title: Preservers of matrix pairs with a fixed inner product value
zbMATH Open document ID: 1255.15033
author: Chi-Kwong Li
author: Lucijan Plevnik
author: Peter Šemrl
published in: Operators and Matrices
publication date: 2012-09-21T00:00:00Z
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 6084687
zbMATH Keywords: Hermitian matrix
zbMATH Keywords: real symmetric matrix
zbMATH Keywords: projection
zbMATH Keywords: effect
zbMATH Keywords: trace
zbMATH Keywords: preserver
MaRDI profile type: Publication
full work available at URL: oam-06-29
OpenAlex ID: W2316245536
DOI: 10.7153/OAM-06-29
review text: The authors characterize bijective maps \(\phi:V\rightarrow V\), where \(V\) is a set of matrices, such that \(\text{tr}(AB)=c \Leftrightarrow \text{tr}\phi(A)\phi(B)=c\) for a fixed \(c\).NEWLINENEWLINEIn Section 1, they give an overall view of the motivation of the article, the connection with quantum physics, and related open questions.NEWLINENEWLINEIn Section 2, they completely characterize the bijections for \(V\) to the set of \(n\times n\) Hermitian matrices or the set of \(n\times n\) symmetric matrices and for arbitrary \(c\).NEWLINENEWLINEIn Section 3, they completely characterize the bijections for \(V\) to be the set of all real effects (i.e matrices \(A\) satisfying \(0\leq A\leq I\)) or the set of all complex effects, and for arbitrary \(c\in [0,1]\).NEWLINENEWLINEIn Section 4, they characterize the bijections for \(V\) to be the set of all projections of rank one and \(c\) lies within \([\frac12, 1)\). It turns out that they are all of the form \(\phi(P)=OPO^t\) for some orthogonal transformation \(O\).
description: scientific article; zbMATH DE number 6084687