Sequential determination of the \(\{1, 4\}\)-inverse of a matrix
type: Item
instance of: scholarly article
title: Sequential determination of the \(\{1, 4\}\)-inverse of a matrix
zbMATH Open document ID: 1040.65033
DOI: 10.1023/A:1023644505061
published in: Journal of Optimization Theory and Applications
publication date: 2003-10-29T00:00:00Z
review text: The authors provide a sequential approach of the \(\{1,4\}\)-generalized inverse of the matrix \(B=[A,a]\), where \(A\) is an \(m\) by \(r\) matrix and \(a\) is an \(m\) by \(p\) block that is appended to it. For that, firstly they give the \(\{1,4\}\)-generalized inverse in terms of the \(\{1,2,4\}\)-generalized inverse of the matrix \(A\). Secondly, they establish the relationship between the \(\{1,4\}\)-generalized inverse of any given matrix and its \(\{1,2,4\}\)-generalized inverse. Finally they show a sequential determination of the \(\{1,4\}\)-generalized inverse of an augmented matrix \(B\) in terms of a \(\{1,4\}\)-generalized inverse of the original matrix \(A\). Moreover, it is proved that if an \(m\) by \(n\) matrix \(A\) has rank \(m\), then its \(\{1,2,4\}\)-generalized inverse is equal to its \(\{1,2,3,4\}\)-generalized inverse. In general, the proofs are rather technical. The authors use the fact that the \(\{1,4\}\)-inverse provides the unique minimum length solution \(x=B^{\{1,4\}}b\), and the Moore-Penrose properties of generalized inverses.
reviewed by: R. Bru
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 1997745
zbMATH Keywords: generalized inverses
zbMATH Keywords: sequential determination
zbMATH Keywords: addition of block-column matrices
zbMATH Keywords: \(\{1, 4\}\)-inverse
zbMATH Keywords: minimum length solution
author: Firdaus E. Udwadia
author: R. E. Kalaba
MaRDI profile type: Publication
cites work: General forms for the recursive determination of generalized inverses: Unified approach
description: scientific article; zbMATH DE number 1997745