Geodesic
On special solutions to the matrix equations $X\bar X=I$ и $X\bar X=-I$
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 9, pp. 1460-1466 
Kh. D. Ikramov. On special solutions to the matrix equations $X\bar X=I$ и $X\bar X=-I$. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 9, pp. 1460-1466. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_9_a1/
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     title = {On special solutions to the matrix equations $X\bar X=I$ {\cyri} $X\bar X=-I$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The matrix equations $X\bar X=I$ and $X\bar X=-I$ are important in the theory of consimilarity. For the first equation, a characterization of solutions was given in Section 4.6 of Matrix Analysis by Horn and Johnson. Since this characterization is not constructive, a complete and constructive description of solutions to these equations is derived under one of the following assumptions: (a) $X$ is a normal matrix, or (b) $X$ is a conjugate-normal matrix.

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

[2] Ikramov Kh. D., “Ob ispolzovanii blochnykh simmetrii pri reshenii algebraicheskikh zadach na sobstvennye znacheniya”, Zh. vychisl. matem. i matem. fiz., 30:11 (1990), 1614–1624 | MR

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