A proof of the following result, due to T. Crimmins, is proposed: A matrix A ∈ M n (C) can be represented as a product of orthoprojectors P and Q if and only if A satisfies the equation A 2 = AA*A. Bibliography: 3 titles.
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References
G. Corach and A. Maestripieri, “Products of orthogonal projections and polar decompositions,” Linear Algebra Appl., 434, 1594–1609 (2011).
H. Radjavi and J. P. Williams, “Products of self-adjoint operators,” Michigan Math. J., 16, 177–185 (1969).
Y. P. Hong and R. A. Horn, “The Jordan canonical form of a product of a Hermitian and a positive semidefinite matrix,” Linear Algebra Appl., 147, 373–386 (1991).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 395, 2011, pp. 75–85.
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Ikramov, K.D. Products of orthoprojectors and a Theorem of Crimmins. J Math Sci 182, 787–792 (2012). https://doi.org/10.1007/s10958-012-0786-3
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DOI: https://doi.org/10.1007/s10958-012-0786-3