Generalized inverses over integral domains
type: Item
instance of: scholarly article
title: Generalized inverses over integral domains
zbMATH Open document ID: 0712.15004
DOI: 10.1016/0024-3795(90)90229-6
author: K. P. S. Bhaskara Rao
author: K. Manjunatha Prasad
author: R. B. Bapat
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: In an earlier paper [Biol. 49, 179--189 (1983; Zbl 0505.15002)] the second author showed that a matrix of rank \(r\) over an integral domain has a generalized inverse if and only if a linear combination of all minors of the matrix is one. He also gave a procedure for constructing a generalized inverse from a linear combination of minors. This paper shows that any reflective generalized inverse can be obtained by this procedure. In a principal ideal domain the procedure gives all generalized inverses. Finally, a rank \(r\) matrix over an integral domain has a Moore-Penrose inverse if and only if the sum of squares of all \(r\times r\) minors is an invertible element of the integral domain. Many papers on generalized inverses over various types of rings have been published during the last twenty years. This paper has only four references.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15B33
zbMATH DE Number: 4171110
zbMATH Keywords: generalized inverse
zbMATH Keywords: principal ideal domain
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: rings
reviewed by: Stephen L. Campbell
MaRDI profile type: Publication
full work available at URL: 0024-3795(90)90229-6
OpenAlex ID: W2070545742
cites work: On Generalized Inverses of Matrices over Principal Ideal Rings
cites work: On generalized inverses of matrices over integral domains
cites work: Q3800195
description: scientific article; zbMATH DE number 4171110