On g-inverses and the nonsingularity of a bordered matrix \(\begin{pmatrix} A&B\\ C&0\end{pmatrix}\)
type: Item
instance of: scholarly article
title: On g-inverses and the nonsingularity of a bordered matrix \(\begin{pmatrix} A&B\\ C&0\end{pmatrix}\)
zbMATH Open document ID: 0699.15002
DOI: 10.1016/0024-3795(90)90245-8
author: Yong-Lin Chen
author: Bingjun Zhou
published in: Linear Algebra and its Applications
publication date: 1990-01-01T00:00:00Z
review text: The matrix \(M=\begin{pmatrix} A & B \\ C & 0\end{pmatrix}\) satisfies the rank additivity conditions if \(\text{rank}M = \text{rank}(A,B)+ \text{rank}(C,0)= \text{rank}(A^ T,C^ T)+ \text{rank}(B^ T,0).\) Let \(G=\begin{pmatrix} C_ 1 & C_ 4 \\ C_ 3 & -C_ 4\end{pmatrix}\) be a g-inverse of M. Necessary and sufficient conditions are given for \(C_ 1\) to be a g-inverse, outer inverse, reflexive inverse, and Moore-Penrose inverse of A. Necessary and sufficient conditions are also given for M to be nonsingular along with explicit expressions for the \(C_ i\) in \(M^{-1}\).
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 4146162
zbMATH Keywords: bordered matrix
zbMATH Keywords: block-matrix
zbMATH Keywords: rank additivity
zbMATH Keywords: g-inverse
zbMATH Keywords: outer inverse
zbMATH Keywords: reflexive inverse
zbMATH Keywords: Moore-Penrose inverse
reviewed by: Stephen L. Campbell
MaRDI profile type: Publication
full work available at URL: 0024-3795(90)90245-8
OpenAlex ID: W1996208176
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description: scientific article; zbMATH DE number 4146162