Geodesic
Unitary similarity of entrywise conjugate matrices and unitary decomplexification
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIII, Tome 382 (2010), pp. 55-59

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Assume that a complex $n\times n$ matrix $A$ is unitarily similar to a real matrix. It is proved that in this case, $A$ is unitarily similar to its entrywise conjugate $\overline A$. An example showing that the reverse implication is in general false is constructed. Bibl. 1 title. 
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     author = {Kh. D. Ikramov},
     title = {Unitary similarity of entrywise conjugate matrices and unitary decomplexification},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {55--59},
     publisher = {mathdoc},
     volume = {382},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_382_a4/}
}
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Kh. D. Ikramov. Unitary similarity of entrywise conjugate matrices and unitary decomplexification. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIII, Tome 382 (2010), pp. 55-59. http://geodesic.mathdoc.fr/item/ZNSL_2010_382_a4/