Geodesic
On sufficient conditions for the existence of a unitary congruence transformation of a given complex matrix into a real one
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIII, Tome 382 (2010), pp. 38-46 
Kh. D. Ikramov. On sufficient conditions for the existence of a unitary congruence transformation of a given complex matrix into a real one. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIII, Tome 382 (2010), pp. 38-46. http://geodesic.mathdoc.fr/item/ZNSL_2010_382_a2/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

A complex $n\times n$ matrix $A$ is said to be nonderogatory if the degree of its minimal polynomial is equal to the degree of the characteristic polynomial. The aim of the paper is to prove the following proposition: Let $A\overline A$ be a nonderogatory matrix with real positive spectrum. Then $A$ can be made real by a unitary congruence transformation if and only if $A$ and $\overline A$ are unitarily congruent. Bibl. 5 titles.

[1] R. Khorn, Ch. Dzhonson, Matrichnyi analiz, Mir, M., 1989 | MR

[2] Kh. D. Ikramov, “A note on complex matrices that are unitarily congruent to real matrices”, Linear Algebra Appl. (to appear) | MR

[3] A. George, Kh. D. Ikramov, E. V. Matushkina, W.-P. Tang, “On a QR-like algorithm for some structured eigenvalue problems”, SIAM J. Matrix Anal. Appl., 16 (1995), 1107–1126 | DOI | MR | Zbl

[4] Y. Hong, R. A. Horn, “A canonical form for matrices under consimilarity”, Linear Algebra Appl., 102 (1988), 143–168 | DOI | MR | Zbl

[5] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1966 | MR