A Rational Criterion for Congruence of Square Matrices

Resumo

With a square complex matrix A the matrix pair consisting of its symmetric S(A) = (A + A^T)/2 and skew-symmetric K(A) = (A − A^T)/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.