congruent matrices

Two matrices are congruent when they are really the same bilinear form seen through two different choices of basis. Just as similarity (P^{-1} A P) is the right equivalence for a linear operator, congruence (P^T A P) is the right equivalence for a bilinear form — because a form's matrix transforms with the transpose, not the inverse, of the change-of-basis matrix.

Precisely, square matrices A and A' over a field K are congruent if there is an invertible matrix P with A' = P^T A P. This is an equivalence relation, and A, A' are congruent exactly when they represent the same bilinear form in two bases related by P. Congruence preserves symmetry (P^T A P is symmetric if A is), the rank, and over R the signature; it does not in general preserve eigenvalues, so congruence is genuinely different from similarity.

The classification of forms is precisely the classification of symmetric matrices up to congruence, and it depends on the field. Over C every nonsingular symmetric matrix is congruent to the identity. Over R the congruence classes are labeled by signature (Sylvester). Over Q the classification is subtle and is the realm of the Hasse-Minkowski theorem, with rank, discriminant modulo squares, and Hasse invariants as the deciding data.