On bordering of regular matrices
type: Item
instance of: scholarly article
title: On bordering of regular matrices
zbMATH Open document ID: 0841.15009
DOI: 10.1016/0024-3795(94)00116-2
author: K. Manjunatha Prasad
author: K. P. S. Bhaskara Rao
published in: Linear Algebra and its Applications
publication date: 1996-03-20T00:00:00Z
review text: \textit{K. Nomakuchi's} characterization of generalized inverses (\(g\)-inverses) of matrices over the field of complex numbers using bordered matrices [ibid. 33, 1--8 (1980; Zbl 0442.15002)] in generalized to matrices over an arbitrary ring. Necessary and sufficient conditions are given for a commutative ring \(R\) to be a ring over which every regular matrix can be completed to an invertible matrix of a particular size by bordering. More precisely, for a regular \(A\), if \(G_{n\times m}\) is a \(g\)-inverse of \(A\), i.e., if \(AGA= A\), then \(\ker(A)= \text{Range}(I-GA)\) and \(\text{coker}(A)= \text{Range}(I-AG)\), and if \(I_n= AG= B_{m\times (m-r)} C_{(m-r)\times m}\) and \(I_m-GA= P_{n\times (n-r)} Q_{(n-r)\times n}\), then \((\begin{smallmatrix} A\\ Q\end{smallmatrix} \begin{smallmatrix} B\\ 0\end{smallmatrix})\) gives an invertible bordering of \(A\). In other words, a regular matrix \(A\) over \(R\) has nomempty set of invertible matrices \[ {\mathcal B}(A)= \{T= (\begin{smallmatrix} A\\ Q\end{smallmatrix} \begin{smallmatrix} P\\ R\end{smallmatrix})\} \] if and only if its kernel and cokernel are free. Such rings are precisely the projective free rings. Also, over such rings every regular matrix has a rank factorization. Using the bordering technique, the authors give an interesting method of computing minors of a reflexive \(g\)-inverse \(G\) of a regular matrix \(A\) when \(I-AG\) and \(I-GA\) have factorizations.
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 13C10
zbMATH DE Number: 847504
zbMATH Keywords: generalized inverses
zbMATH Keywords: bordered matrices
zbMATH Keywords: invertible matrix
zbMATH Keywords: projective free rings
zbMATH Keywords: regular matrix
zbMATH Keywords: rank factorization
zbMATH Keywords: reflexive \(g\)-inverse
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)00116-2
OpenAlex ID: W2067313914
cites work: Generalized inverses over integral domains
cites work: Q4061081
cites work: On generalized inverses of matrices over integral domains
cites work: On Generalized Inverses of Matrices over Principal Ideal Rings
cites work: Bordered Matrices
cites work: Q3254327
cites work: Weak generalized inverses and minimum variance linear unbiased estimation
cites work: Q4771385
cites work: Reflexive generalized inverses and their minors*
cites work: On the characterization of generalized inverses by bordered matrices
cites work: Projective modules over polynomial rings
cites work: On generalized inverses of polynomial and other matrices
Recommended article: Q2784581
Recommended article: Generalized inverses over integral domains
Recommended article: Reflexive generalized inverses and their minors*
Recommended article: Generalized inverses of bordered matrices
Recommended article: Q4025694
Recommended article: Q3795792
Recommended article: Rank and dimension functions
Recommended article: Generalized Hirano inverses in rings
Recommended article: Q3317237
Recommended article: Q3743427
description: scientific article; zbMATH DE number 847504