Categories of matrices with only obvious Moore-Penrose inverses
type: Item
instance of: scholarly article
title: Categories of matrices with only obvious Moore-Penrose inverses
zbMATH Open document ID: 0633.15002
DOI: 10.1016/0024-3795(87)90142-X
author: Donald W. Robinson
author: R. Puystjens
author: Jan Van Geel
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let R be an associative ring with 1 and with an involution \(a\to \bar a\), and let \(M_ R\) be the category of finite matrices over R with the involution \((a_{ij})\to (a_{ij})^*=(\bar a_{ji})\). The authors prove that the following two statements are equivalent: (1) If A in \(M_ R\) has a Moore-Penrose inverse with respect to *, then A is permutationally equivalent to a matrix of the form \(\left( \begin{matrix} B\quad 0\\ 0\quad 0\end{matrix} \right)\) with B invertible. (2) If \(1=\Sigma a\bar a\) in R, then at most one of the a's is not zero. Some other equivalent conditions are also contained.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 4029753
zbMATH Keywords: associative ring
zbMATH Keywords: involution
zbMATH Keywords: finite matrices
zbMATH Keywords: Moore-Penrose inverse
reviewed by: Shaokuan Li
MaRDI profile type: Publication
cites work: Integral generalized inverses of integral matrices
cites work: Q5639812
cites work: The Moore-Penrose inverse of a morphism with factorization
cites work: On generalized inverses of matrices over integral domains
full work available at URL: 0024-3795(87)90142-x
OpenAlex ID: W2089591814
description: scientific article; zbMATH DE number 4029753