Geodesic
Products of orthoprojectors and Hermitian matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 67-70 Cet article a éte moissonné depuis la source Math-Net.Ru

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A proof of the following result is presented: A matrix $A\in M_n(\mathbf C)$ can be represented as a product $A=PH$, where $P$ is an orthoprojector and $H$ is Hermitian, if and only if $A$ satisfies the equation $A^{*2}A=A^*A^2$ (the Radjavi–Williams theorem). Unlike the original proof, ours makes no use of the Crimmins theorem. 
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     author = {Kh. D. Ikramov},
     title = {Products of orthoprojectors and {Hermitian} matrices},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a5/}
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Kh. D. Ikramov. Products of orthoprojectors and Hermitian matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 67-70. http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a5/

[1] H. Radjavi, J. P. Williams, “Products of self-adjoint operators”, Michigan Math. J., 16 (1969), 177–185. | DOI | MR | Zbl

[2] Y. P. Hong, R. A. Horn, “The Jordan canonical form of a product of a Hermitian and a positive semidefinite matrix”, Linear Algebra Appl., 147 (1991), 373–386 | DOI | MR | Zbl