Geodesic
A criterion for unitary congruence between complex matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 9-19

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Let $A$ and $B$ be square complex matrices. Based on an important result of Y. P. Hong and R. A. Horn, we propose a criterion for verifying unitary congruence of these matrices. The criterion requires that a finite number of arithmetic operations be performed. No criteria with this finiteness property were previously known. 
@article{ZNSL_2011_395_a1,
     author = {Yu. A. Al'pin and Kh. D. Ikramov},
     title = {A criterion for unitary congruence between complex matrices},
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     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a1/}
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Yu. A. Al'pin; Kh. D. Ikramov. A criterion for unitary congruence between complex matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXIV, Tome 395 (2011), pp. 9-19. http://geodesic.mathdoc.fr/item/ZNSL_2011_395_a1/