signature
For a real form, diagonalizing it sorts the directions into three bins: those on which the form is positive, those on which it is negative, and those on which it is zero. The signature is the headcount of the first two bins. It is the fingerprint that tells, for instance, the difference between Euclidean geometry (all positive) and the spacetime of relativity (mostly positive, one negative).
Over R, diagonalize a quadratic form Q to a_1 y_1^2 + … + a_n y_n^2; let p be the number of strictly positive a_i and q the number of strictly negative ones. The signature is the pair (p, q), sometimes reported as the single integer p - q or written p − q. Sylvester's law of inertia guarantees p and q do not depend on which diagonalizing basis was used, so the signature is a genuine invariant.
The rank is p + q and the nullity is n − (p + q). The form is positive definite when q = 0 and p = n, negative definite when p = 0 and q = n, and indefinite when both p and q are positive. Two real quadratic forms are congruent if and only if they have the same dimension and the same signature — so over R the signature, together with rank, completely classifies forms.
The Minkowski form Q(t, x, y, z) = -t^2 + x^2 + y^2 + z^2 has signature (3, 1): three positive directions and one negative, rank 4, nondegenerate but indefinite.
The signature of spacetime's metric form.
The notion of signature is special to ordered fields like R. Over C every nondegenerate form of a given rank is equivalent (one cannot distinguish +1 from -1 because i^2 = -1), so no analogue of signature survives.