The determinal rank idempotents of a matrix
type: Item
instance of: scholarly article
title: The determinal rank idempotents of a matrix
zbMATH Open document ID: 0848.15004
DOI: 10.1016/0024-3795(95)00450-5
author: Donald W. Robinson
published in: Linear Algebra and its Applications
publication date: 1996-10-09T00:00:00Z
review text: Let \({\mathbf M}_R\) be the category of finite matrices over a commutative ring \(R\) with 1. For \(A\in {\mathbf M}_R\) of determinantal rank \(r\) let \({\mathcal I}_r (A)\) be the ideal of \(R\) generated by the \(r\times r\) minors of \(A\). There exists a unique list \((e_1, \dots, e_t)\) of pairwise orthogonal idempotents of \(R\) that sum to 1 such that, if \(r_i= \text{rank} (e_i A)\), then \(\text{rank } A= r_1> r_2> \dots> r_t\), \(e_i\) is the identity element of \({\mathcal I}_{r_i} (e_i A)\) for \(1\leq i<t\), and either \(e_t A=0\) or \({\mathcal I}_{r_t} (e_t A)\) does not possess an identity element. Characterizations are given for various generalized inverses of \(A= e_1 A+ \dots+ e_t A\). In particular, \(A\) is regular if and only if \(e_t A=0\).
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 885060
zbMATH Keywords: pairwise orthogonality
zbMATH Keywords: finite matrices
zbMATH Keywords: commutative ring
zbMATH Keywords: determinantal rank
zbMATH Keywords: ideal
zbMATH Keywords: minors
zbMATH Keywords: pairwise orthogonal idempotents
zbMATH Keywords: generalized inverses
reviewed by: Giacomo Bonanno
MaRDI profile type: Publication
full work available at URL: 0024-3795(95)00450-5
OpenAlex ID: W1984838303
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description: scientific article; zbMATH DE number 885060