The signature (or inertia) of a symmetric bilinear form or quadratic form over the reals is the ordered triple (p, n, z) giving the numbers of positive, negative, and zero eigenvalues of any representing symmetric matrix, describing its definiteness pattern.

Principle

Sylvester's law of inertia: under congruence transformations over R the counts p and n are invariant, so signature classifies quadratic forms up to real change of variables; it partitions forms into definite, semidefinite, and indefinite types.

Demonstration

A symmetric 3×3 matrix diagonalized to diag(2, -1, 0) has signature (1,1,1): one positive eigenvalue, one negative, one zero. This determines that the associated quadratic form is indefinite and singular.

Misapplication

Using determinant sign or rank as a substitute for signature; for example, assuming a positive determinant implies positive definiteness ignores possible negative eigenvalues and zero eigenvalues.

Consequence

Knowing the signature lets one decide definiteness properties, count positive and negative directions for optimization and Morse theory, and determine the qualitative geometry of level sets of the quadratic form.

Reversal

Reversing signs of the form (multiplying by −1) swaps positive and negative counts, turning positive definite into negative definite; conceptually, the reversal highlights complementary directions but preserves the number of zero modes.

Boundary

Defined primarily for symmetric bilinear or real quadratic forms (and Hermitian forms with adaptations) over fields where ordering makes sense; over algebraically closed fields or without a real order, signature is not meaningful in the same way.

Semantic Tension

Tension arises between signature and invariants like determinant, rank, or eigenvalue lists: these are related but differ in granularity or invariance under congruence versus similarity, so they can disagree in classification power.

Synthesis

Signature is the congruence-invariant count of positive, negative, and zero spectral directions of a real symmetric form, providing the decisive classification for definiteness and the qualitative shape of the associated quadratic geometry.