In mathematics, the signature (v, p, r)[clarification needed] of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis.
Alternatively, it can be defined as the dimensions of a maximal positive and null subspace.
By Sylvester's law of inertia these numbers do not depend on the choice of basis and thus can be used to classify the metric.
There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as (v − p), where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit.
[2] It is the number (v, p, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities.
By Sylvester's law of inertia, the numbers (v, p, r) are basis independent.
real symmetric bilinear form), g does not depend on the choice of basis.
Likewise the signature is equal for two congruent matrices and classifies a matrix up to congruency.
Equivalently, the signature is constant on the orbits of the general linear group GL(V) on the space of symmetric rank 2 contravariant tensors S2V∗ and classifies each orbit.
p) is the maximal dimension of a vector subspace on which the scalar product g is positive-definite (resp.
The following matrices have both the same signature (1, 1, 0), therefore they are congruent because of Sylvester's law of inertia: The standard scalar product defined on
The signature counts how many time-like or space-like characters are in the spacetime, in the sense defined by special relativity: as used in particle physics, the metric has an eigenvalue on the time-like subspace, and its mirroring eigenvalue on the space-like subspace.
(Sometimes the opposite sign convention is used, but with the one given here s directly measures proper time.)
[3] Such signature changing metrics may possibly have applications in cosmology and quantum gravity.