Rank equalities related to the generalized inverses \(A_{T,S}^{(2)}\), \(B_{T_1,S_1}^{(2)}\) of two matrices \(A\) and \(B\)
type: Item
instance of: scholarly article
title: Rank equalities related to the generalized inverses \(A_{T,S}^{(2)}\), \(B_{T_1,S_1}^{(2)}\) of two matrices \(A\) and \(B\)
zbMATH Open document ID: 1063.15003
author: Musheng Wei
author: Yonghui Liu
published in: Applied Mathematics and Computation
publication date: 2005-01-17T00:00:00Z
review text: Let \(A\) be an \(m\times n\) matrix over \(\mathbb{C}\). A \{2\}-inverse of \(A\) is an \(n\times m\) matrix \(X\) such that \(XAX=X\) [see \textit{A. Ben-Israel} and \textit{T. N. E. Greville}, Generalized inverses: Theory and applications (Wiley, New York) (1974; Zbl 0305.15001) for background on generalized matrix inverses]. Let \(T\) and \(S\) be subspaces of \(\mathbb{C}^{n}\) and \(\mathbb{C}^{m}\), respectively, with \(m-\dim S=\dim T\leq rank(A)\). Then there exists a unique \{2\}-inverse \(X\) of \(A\) with range \(T\) and null space \(S\) if and only if \(AT\oplus S=\mathbb{C}^{m}\); this inverse is denoted \(A_{T,S}^{(2)}\). More explicitly, if \(G\) is any \(n\times m\) matrix with range \(T\) and null space \(S\), and \((AG)_{g}\) denotes the group inverse of \(AG\), then \(A_{T,S}^{(2)}=G(AG)_{g}\). For specific choices of \(S\) and \(T\), \(A_{T,S}^{(2)}\) is equal to the Moore-Penrose inverse, the group inverse, the Drazin inverse, and various other generalized inverses. The authors prove some inequalities relating to the ranks of matrices and their generalized inverses. The following is typical. Let \(A\) and \(B\) be matrices of dimensions \(m\times n\) and \(m\times k\), respectively, and \(G_{a}\) and \(G_{b}\) be of dimensions \(n\times m\) and \(k\times m\), respectively. Suppose that the range and null space of \(G_{a}\) (respectively \(G_{b}\)) are \(T\) and \(S\) (\(T_{1}\) and \(S_{1}\)) and that the generalized inverses \(A_{T,S}^{(2)}\) and \(B_{T_{1},S_{1}}^{(2)}\) exist. Then \(rank(AA_{T,S} ^{(2)}-BB_{T_{1},S_{1}}^{(2)})=rank(G_{a}^{T},G_{b}^{T})+rank(AG_{a} ,BG_{b})-rank(G_{a})-rank(G_{b})\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 2128787
zbMATH Keywords: generalized inverse \(A_{T,S}^{(2)}\)
zbMATH Keywords: rank
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: Drazin inverse
zbMATH Keywords: group inverse
zbMATH Keywords: \{2\}-inverse
zbMATH Keywords: inequalities
MaRDI profile type: Publication
full work available at URL: j.amc.2003.08.124
OpenAlex ID: W2049141741
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DOI: 10.1016/J.AMC.2003.08.124
description: scientific article; zbMATH DE number 2128787