Geodesic
Products of orthoprojectors and Hermitian matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 67-70

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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. 
@article{ZNSL_2011_395_a5,
     author = {Kh. D. Ikramov},
     title = {Products of orthoprojectors and {Hermitian} matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {67--70},
     publisher = {mathdoc},
     volume = {395},
     year = {2011},
     language = {ru},
     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/