Sylvester's law of inertia

You can diagonalize a real quadratic form in many ways, and the actual numbers on the diagonal will vary from one method to another. Sylvester's law of inertia says one thing nonetheless stays fixed: how many of those numbers are positive and how many are negative. No clever choice of coordinates can convert a positive direction into a negative one. The signature is therefore not an artifact of bookkeeping but a real property of the form.

Statement: if a real symmetric bilinear form (equivalently a real quadratic form) is brought to diagonal shape in two different bases, the number p of positive diagonal entries and the number q of negative entries are the same in both. Equivalently, the maximal dimension of a subspace on which Q is positive definite equals p, and on which it is negative definite equals q; these dimensions are basis-free, which forces the diagonal counts to agree.

The result was published by Sylvester in 1852 and is the cornerstone of the real classification: it makes (p, q) a well-defined invariant, so two real forms are congruent precisely when they share dimension and signature. The proof typically argues that a positive-definite subspace and the span of the negative and null diagonal directions must intersect trivially, bounding p, and symmetrically bounding q. The law fails over C (where signs can be absorbed) and over fields with no order.