A note on an algebraic version of Cochran's theorem
type: Item
instance of: scholarly article
title: A note on an algebraic version of Cochran's theorem
zbMATH Open document ID: 1059.15018
author: Gorazd Lešnjak
published in: Linear Algebra and its Applications
publication date: 2004-10-04T00:00:00Z
review text: Let \(M_n\) be the algebra of all complex \(n \times n\) matrices, \(n > 1\). The algebraic version of \textit{W. G. Cochran}'s theorem [Proc. Cambridge Philos. Soc. 30, 178--191 (1934; Zbl 0009.12004)] in statistics is linking the notions of orthogonal and rank additive family of matrices in \(M_n\): (i) A rank additive decomposition of an idempotent matrix is orthogonal. (ii) An orthogonal family of idempotent matrices is rank additive. The first part was extended before by \textit{P. Šemrl} [Linear Algebra Appl. 237--238, 477--487 (1996; Zbl 0847.62043)] for all matrices \(A\) having the property that every rank additive family with sum \(A\) is orthogonal. The author concludes this generalization extending the second part, i.e. characterizing all square matrices \(A\) for which every orthogonal family of matrices having \(A\) as its sum is rank additive.
reviewed by: Daniela B. Nikolova
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 2105749
zbMATH Keywords: rank additive matrix family
zbMATH Keywords: orthogonal matrix family
zbMATH Keywords: Cochran's theorem
MaRDI profile type: Publication
full work available at URL: j.laa.2004.02.033
OpenAlex ID: W2160699246
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DOI: 10.1016/J.LAA.2004.02.033
description: scientific article; zbMATH DE number 2105749