congruence

Two square matrices B and C are congruent if C = P^T B P for some invertible matrix P. This is the right notion of 'the same' for bilinear forms: B and P^T B P represent the very same form, just written in two different bases. Congruence is the form-world counterpart of similarity, which (with P^-1 instead of P^T) is the right notion for operators.

The reason congruence is the correct relation comes straight from the change-of-basis law. If you re-express vectors by x = P x' (new coordinates x'), then x^T B x = (P x')^T B (P x') = x'^T (P^T B P) x'. So the matrix of the form in the new basis is exactly P^T B P. Anything you call a property 'of the form' must therefore be a congruence invariant, unchanged when you replace B by P^T B P.

Congruence is an equivalence relation — reflexive, symmetric, transitive — so it carves all symmetric matrices into classes, and classifying the classes is the goal of the field. Over the complex numbers a symmetric matrix is congruent to a matrix that is the identity in its first r diagonal slots and zero afterward, so rank alone is the complete invariant. Over the reals you need rank and the signs: that is Sylvester's law of inertia.

Do not confuse congruence with similarity. Eigenvalues are similarity invariants but they are NOT congruence invariants — congruence can change the eigenvalues of a symmetric matrix while preserving only the SIGNS of the eigenvalues. What survives congruence is exactly the signature, not the spectrum. Mixing the two relations is the most common Vol II error in this whole chapter.