Every matrix is a linear combination of three idempotents
type: Item
instance of: scholarly article
title: Every matrix is a linear combination of three idempotents
zbMATH Open document ID: 1058.15014
author: Vyacheslav Rabanovich
published in: Linear Algebra and its Applications
publication date: 2004-11-18T00:00:00Z
review text: It is shown that every matrix \(A\in \mathbb{F}^{n\times n}\) over a field \(\mathbb{F}\) of characteristic zero may be represented as \(\lambda_1 Q_1+ \lambda_2 Q_2+ \lambda_3 Q_2\), where \(\lambda_i\in \mathbb{F}\), \(Q_i\in \mathbb{F}^{n\times n}\) and \(Q^2_i= Q_i\). If \(A\in \mathbb{F}^{2\times 2}\) or \(A\in \mathbb{C}^{3\times 3}\) it is fulfilled \(A= \lambda_1 Q_1+ \lambda_2 Q_2\). At the same time there are matrices \(A\in \mathbb{Q}^{3\times 3}\) and \(A\in \mathbb{C}^{4\times 4}\) that are not linear combinations of two idempotents.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A23
zbMATH DE Number: 2116528
zbMATH Keywords: Idempotent
zbMATH Keywords: Linear combination
zbMATH Keywords: Decomposition
reviewed by: Mihail M. Konstantinov
MaRDI profile type: Publication
full work available at URL: j.laa.2004.04.014
OpenAlex ID: W2018075733
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cites work: Q4308114
DOI: 10.1016/J.LAA.2004.04.014
description: scientific article; zbMATH DE number 2116528