From projectors to 1MP and MP1 generalized inverses and their induced partial orders
type: Item
instance of: scholarly article
title: From projectors to 1MP and MP1 generalized inverses and their induced partial orders
zbMATH Open document ID: 1490.15008
author: M. V. Hernández
author: Néstor Thome
author: M. Lattanzi
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2021-08-30T00:00:00Z
review text: Let \(A\) be an \(m \times n\) complex matrix. For each \(A^{ - } \in \mathcal{A} \{1 \}\), the matrix \[A^{ - \, \dag }:= A^{ - } A A^{ \dag } \in \mathbb{C}^{n \times m}\] is called a 1MP-inverse of \(A\). For each \(A^{ - } \in \mathcal{A} \{1 \}\), the MP1-inverse of \(A\), denoted by \(A^{ \dag \, - }\), is the \(n \times m\) matrix \[A^{ \dag \, - }:= A^{ \dag } A A^{ - }.\] This paper is devoted to the study of the above generalized inverses of rectangular complex matrices, 1MP and MP1-inverses. The 1MP-inverses of a matrix \(A\) are characterized from a singular value decomposition of \(A\). The authors give some characterizations of 1MP-inverses of a matrix \(A\) as \(\{1, 2, 3 \}\)-inverses of \(A\), as the solutions of a matrix equation system, and as a 1-parametrized set. As an application, they analyze the restriction of 1MP-inverses to the set of partial isometries. They also introduce and study a partial order associated to 1MP-inverses as other application of 1MP-inverses. Indeed, the binary relations induced by these new generalized inverses are proved to be partial orders. The dual case (called MP1-inverse) is studied too.
reviewed by: Kamran Sharifi
Mathematics Subject Classification ID: 15A10
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 7388096
zbMATH Keywords: generalized inverses
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: matrix equations
zbMATH Keywords: partial order
zbMATH Keywords: equivalence classes
MaRDI profile type: Publication
full work available at URL: s13398-021-01090-8
OpenAlex ID: W3174428553
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DOI: 10.1007/S13398-021-01090-8
description: scientific article; zbMATH DE number 7388096