Geodesic
How to check whether given square matrices are congruent?
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVIII, Tome 439 (2015), pp. 99-106 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $A$ and $B$ be square nonsingular $n$-by-$n$ matrices with entries being rational or rational Gaussian numbers. We describe a method for verifying whether these matrices are congruent. The method uses a finite number of arithmetic operations (and, in the complex case, conjugation operations). 
@article{ZNSL_2015_439_a8,
     author = {Kh. D. Ikramov},
     title = {How to check whether given square matrices are congruent?},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {99--106},
     year = {2015},
     volume = {439},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_439_a8/}
}
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Kh. D. Ikramov. How to check whether given square matrices are congruent?. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXVIII, Tome 439 (2015), pp. 99-106. http://geodesic.mathdoc.fr/item/ZNSL_2015_439_a8/