Further results on the reverse order law for \(\{1,3\}\)-inverse and \(\{1,4\}\)-inverse of a matrix product
type: Item
instance of: scholarly article
title: Further results on the reverse order law for \(\{1,3\}\)-inverse and \(\{1,4\}\)-inverse of a matrix product
zbMATH Open document ID: 1207.15005
DOI: 10.1155/2010/312767
author: Deqiang Liu
author: Hu Yang
published in: Journal of Inequalities and Applications
publication date: 2010-12-06T00:00:00Z
review text: For a complex \(m \times n\) matrix \(A\), a generalized inverse \(X\) of \(A\) is a matrix, of size \(n \times m\), which satisfies some of the following equations: \[ (1)\;AXA=A, \;\;(2) \;XAX=X, \;\;(3) \;(AX)^*=AX, \;\;(4) \;(XA)^*=XA, \] where \(A^*\) denotes the conjugate transpose of \(A\). Let \(\eta\) be a nonempty subset of \(\{1,2,3,4\}\). In this paper, \(A \eta\) denotes the set of all matrices \(X\) which satisfy (i) for all \(i \in \eta\). Any matrix \(X \in A \eta\) is called an \(\eta\)-inverse of \(A\). One usually denotes any \(\{1\}\)-inverse of \(A\) by \(A^{(1)}\) or \(A^{-}\), and any \(\{1,3\}\)-inverse of \(A\) by \(A^{(1,3)}\) which is also called a least squares \(g\)-inverse of \(A\). Any \(\{1,4\}\)-inverse of \(A\) is denoted by \(A^{(1,4)}\) which is also called a minimum norm \(g\)-inverse of \(A\). The unique \(\{1,2,3,4\}\)-inverse of \(A\) is called the Moore-Penrose generalized inverse of \(A\). Firstly, in this work the authors give the minimal rank of \(D-B^{(1,3)}A^{(1,3)}\) with respect to any \(B^{(1,3)}\) and \(A^{(1,3)}\), being \(A\), \(B\) and \(D\) complex matrices of size \(m \times n\), \(n \times k\) and \(k \times m\), respectively. Secondly, the authors present necessary and sufficient conditions for the inclusion \(B\{1,3\}A\{1,3\}\supseteq (AB)\{1,3\}\). Specifically, they prove that the following statements are equivalent: {\parindent=6mm \begin{itemize}\item[(a)] \(B\{1,3\}A\{1,3\}\supseteq (AB)\{1,3\}\), \item[(b)] \(r(A^*AB,B)+r(A)=r(AB)+\min \{ r(A^*,B), \max \{n+r(A)-m,n+r(B)-k \} \}\), \end{itemize}} where \(r(A)\) denotes the rank of the matrix \(A\). Finally, the authors show necessary and sufficient conditions for \(B\{1,3\}A\{1,3\}=(AB)\{1,3\}\), \(B\{1,4\}A\{1,4\}\supseteq (AB)\{1,4\}\) and \(B\{1,4\}A\{1,4\}= (AB)\{1,4\}\).
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 5823161
zbMATH Keywords: generalized inverse
zbMATH Keywords: g-inverse
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: range space
zbMATH Keywords: null space
zbMATH Keywords: reverse order law
zbMATH Keywords: matrix product
zbMATH Keywords: minimal rank
Wikidata QID: Q59253953
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2010/312767
OpenAlex ID: W2088593928
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description: scientific article; zbMATH DE number 5823161