Further properties of generalized and hypergeneralized projectors
type: Item
instance of: scholarly article
title: Further properties of generalized and hypergeneralized projectors
zbMATH Open document ID: 1068.15036
author: Jerzy K. Baksalary
author: Oskar Maria Baksalary
author: Xiaoji Liu
published in: Linear Algebra and its Applications
publication date: 2004-11-22T00:00:00Z
review text: It is well known that a complex \(n\times n\) matrix \(K\) is a projector on the vector space \({\mathbb C}^{n}\) precisely when \(K^2\) equals the identity matrix \(I_ n\). Moreover, a projector is orthogonal (in terms of the standard unitary scalar product) if, and only if, \(K\) coincides with its conjugate transpose \(K^*\). Following \textit{J.~Groß} and \textit{G.~Trenkler} [Linear Algebra Appl. 264, 463-474 (1997; Zbl 0887.15024)], the authors consider \textit{generalized} and \textit{hypergeneralized} projectors; they are defined via the conditions \(K^ 2= K^ *\) and \(K^ 2 = K^ +\) (the Moore-Penrose inverse of \(K\)), respectively. Several properties of such matrices are established. Some of the results obtained extend and/or generalize those given in the reference above, whereas others are solutions to new problems. For example, the following holds: \(K\) is a (hyper)generalized projector if, and only if, \(K^ +\) shares the same property.
reviewed by: Hans Havlicek
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 2117299
zbMATH Keywords: generalized projector
zbMATH Keywords: hypergeneralized projector
zbMATH Keywords: idempotent matrix
zbMATH Keywords: quadripotent matrix
zbMATH Keywords: partial isometry
zbMATH Keywords: EP matrix
zbMATH Keywords: normal matrix
zbMATH Keywords: Moore-Penrose inverse
MaRDI profile type: Publication
full work available at URL: j.laa.2004.03.013
OpenAlex ID: W1972497290
cites work: Further properties of the star, left-star, right-star, and minus partial orderings.
cites work: A property of orthogonal projectors
cites work: An alternative characterization of generalized projectors
cites work: Generalized and hypergeneralized projectors
DOI: 10.1016/J.LAA.2004.03.013
description: scientific article; zbMATH DE number 2117299