Geodesic
A rational criterion for congruence of square matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXI, Tome 472 (2018), pp. 88-91 
Kh. D. Ikramov. A rational criterion for congruence of square matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXI, Tome 472 (2018), pp. 88-91. http://geodesic.mathdoc.fr/item/ZNSL_2018_472_a5/
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     author = {Kh. D. Ikramov},
     title = {A rational criterion for congruence of square matrices},
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     pages = {88--91},
     year = {2018},
     volume = {472},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2018_472_a5/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

With a square complex matrix $A$ we associate the matrix pair consisting of its symmetric part $S(A) = (A + A^T)/2$ and its skew-symmetric part $K(A) = (A - A^T)/2$. We show that square matrices $A$ and $B$ are congruent if and only if the associated pairs $(S(A),K(A))$ and $(S(B),K(B))$ are (strictly) equivalent. This criterion can be verified by a finite rational calculation if the entries of $A$ and $B$ are rational or rational Gaussian numbers.

[1] R. A. Horn, C. R. Johnson, Matrix Analysis, Second Edition, Cambridge University Press, 2013 | MR | Zbl

[2] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1966

[3] Kh. D. Ikramov, “Vydelenie regulyarnoi chasti singulyarnogo matrichnogo puchka kak ratsionalnyi algoritm”, Zap. nauchn. semin. POMI, 439, 2015, 107–111