On the equality between rank and trace of an idempotent matrix
type: Item
instance of: scholarly article
title: On the equality between rank and trace of an idempotent matrix
zbMATH Open document ID: 1206.15006
author: Oskar Maria Baksalary
author: Dennis S. Bernstein
author: Götz Trenkler
published in: Applied Mathematics and Computation
publication date: 2011-01-14T00:00:00Z
review text: The paper deals with the question whether it is possible to derive the equality between the rank and trace of an idempotent matrix by using only the idempotency property, without referring to any further features of the matrix. It is shown that such a proof can be obtained by exploiting a general characteristic of the rank of any matrix. An original proof of this characteristic is provided, which utilizes a formula for the Moore-Penrose inverse of a partitioned matrix. Further consequences of the rank property are discussed, in particular, several additional facts are established with considerably simpler proofs than those available. Moreover, a collection of new results referring to the coincidence between rank and trace of an idempotent matrix are derived as well.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5836701
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: oblique projector
zbMATH Keywords: orthogonal projector
zbMATH Keywords: partitioned matrix
zbMATH Keywords: rank
zbMATH Keywords: trace
zbMATH Keywords: idempotent matrix
reviewed by: Andreas Arvanitoyeorgos
MaRDI profile type: Publication
full work available at URL: j.amc.2010.10.022
OpenAlex ID: W1988065212
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DOI: 10.1016/J.AMC.2010.10.022
description: scientific article; zbMATH DE number 5836701