A property of orthogonal projectors

It is shown that if P 1 and P 2 are orthogonal projectors, then a product having P 1 and P 2 as its factors is equal to another such product if and only if P 1 and P 2 commute, in which case all products involving P 1 and P 2 reduce to the orthogonal projector P 1 P 2 . This is a generalization of a...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 354; no. 1; pp. 35 - 39
Main Authors Baksalary, Jerzy K., Maria Baksalary, Oskar, Szulc, Tomasz
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.10.2002
Subjects
Commutativity
Hermitian idempotent matrix
Power of a matrix
Product of projectors
Commutativity
Product of projectors
15A27
15A57
Hermitian idempotent matrix
Power of a matrix
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Summary:It is shown that if P 1 and P 2 are orthogonal projectors, then a product having P 1 and P 2 as its factors is equal to another such product if and only if P 1 and P 2 commute, in which case all products involving P 1 and P 2 reduce to the orthogonal projector P 1 P 2 . This is a generalization of a result by Baksalary and Baksalary [Linear Algebra Appl. 341 (2002) 129], with the proof based on a simple property of powers of Hermitian nonnegative definite matrices.
ISSN:0024-3795
1873-1856
DOI:10.1016/S0024-3795(02)00337-3