Generalized inverses of morphisms with kernels
type: Item
instance of: scholarly article
title: Generalized inverses of morphisms with kernels
zbMATH Open document ID: 0633.15001
DOI: 10.1016/0024-3795(87)90336-3
author: Donald W. Robinson
author: R. Puystjens
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let \(\phi: X\to Y\) be a morphism with kernel \(\kappa\) : \(K\to X\) in an additive category with involution *. Then \(\phi\) has a Moore-Penrose inverse \(\phi^{\pi}\) with respect to * iff \(\phi\phi^* + \kappa^*\kappa\) is invertible; in this case, \(\phi^{\pi} = \phi^*(\phi\phi^* + \kappa^*\kappa)^{-1}\). If \(X=Y\), then \(\phi\) has a group inverse \(\phi^{\#}\) iff \(\phi\) has a cokernel \(\gamma: X\to K\) and \(\phi^ 2 + \gamma\kappa\) is invertible. In this case, \(\phi^{\#} = \phi(\phi^ 2 + \gamma\kappa)^{-1}\).
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 18E05
zbMATH DE Number: 4029752
zbMATH Keywords: generalized inverse
zbMATH Keywords: morphism
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: group inverse
reviewed by: Stephen L. Campbell
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90336-3
OpenAlex ID: W2086671596
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description: scientific article; zbMATH DE number 4029752