Revisitation of the product of two orthogonal projectors |
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Authors: | Oskar Maria Baksalary Götz Trenkler |
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Institution: | a Faculty of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614 Poznań, Poland b Department of Statistics, Dortmund University of Technology, Vogelpothsweg 87, D-44221 Dortmund, Germany |
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Abstract: | Several results involving a product of two orthogonal projectors (i.e., Hermitian idempotent matrices) are established by exploring a representation of the product as a partitioned matrix. These results concern, for instance, rank, trace, range, null space, generalized inverses, and spectral properties of the product and its various functions. Particular attention is paid to the conditions equivalent to the requirement that the product of two orthogonal projectors is an orthogonal projector itself, and these characterizations refer to such known classes of matrices as Hermitian, involutory, normal, star-dagger, unitary as well as partial isometries and semi-orthogonal projectors. Moreover, some results dealing with the notions of parallel sum and spectral norm are obtained. The variety of problems considered shows that the approach utilized in the paper provides a powerful tool of wide applicability. |
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Keywords: | 15A57 15A18 15A24 |
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