On generalized inverses of a block in a partitioned matrix
type: Item
instance of: scholarly article
title: On generalized inverses of a block in a partitioned matrix
zbMATH Open document ID: 0655.15005
DOI: 10.1016/0024-3795(88)90203-0
author: P. Bhimasankaram
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: Let M and G be partitioned square matrices consisting of the blocks, A, x, u', c, and B, y, v', d, respectively; c and d denote scalars, x, y, u and v are vectors of appropriate size, and the prime denotes transpose. Provided that G is a generalized inverse of M the author gives a set of conditions for B to be a generalized inverse of A with some specific properties (e.g. \(B=BAB\), AB is symmetric, B is the Moore-Penrose inverse of A). Computational formulae for different generalized inverses of A are also derived. Particular attention is paid to the cases where M and G are symmetric \((x=u\), \(y=v)\) or nonnegative definite. The results can be applied to re-estimation problems of parameters \(\beta =(\beta_ 1,\beta_ 2,...,\beta_ m)'\) of a linear regression model \(Y=X\beta +\epsilon\), \(E(\epsilon)=0\), \(Var(\epsilon)=\sigma\) 2V when one of the parameters is to be dropped.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 62J05
zbMATH DE Number: 4069079
zbMATH Keywords: symmetric and nonnegative definite matrices
zbMATH Keywords: generalized inverse
zbMATH Keywords: Moore- Penrose inverse
zbMATH Keywords: linear regression
reviewed by: László Mihályffy
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90203-0
OpenAlex ID: W2059669101
cites work: Q5638446
cites work: A Generalization of the Schur Complement by Means of the Moore–Penrose Inverse
cites work: Generalized Inverse Formulas Using the Schur Complement
cites work: Q5645536
cites work: Generalized Inverses of Partitioned Matrices
description: scientific article; zbMATH DE number 4069079