With a square complex matrix A the matrix pair consisting of its symmetric S(A) = (A + AT)/2 and skew-symmetric K(A) = (A − AT)/2 parts is associated. It is shown 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 rational calculation, provided that the entries of A and B are rational or rational Gaussian numbers.
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References
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Kh. D. Ikramov, “Isolation of the regular part of a singular matrix pencil as a rational algorithm,” Zap. Nauchn. Semin. POMI, 439, 107–111 (2015).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 472, 2018, pp. 88–91.
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Ikramov, K.D. A Rational Criterion for Congruence of Square Matrices. J Math Sci 240, 762–764 (2019). https://doi.org/10.1007/s10958-019-04392-w
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DOI: https://doi.org/10.1007/s10958-019-04392-w