Метрикалық тензордың өзіндік мәндерінің саны және таңбасы

Number of positive, negative and zero eigenvalues of a metric tensor
In mathematics, the signature (v, p, r) 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. In relativistic physics, v conventionally represents the number of time or virtual dimensions, and p the number of space or physical dimensions. 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. The signature is often denoted by a pair of integers (v, p) implying r = 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3, 1, 0), respectively. The signature is said to be indefinite or mixed if both v and p are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a positive definite signature (v, 0). A Lorentzian metric is a metric with signature (p, 1), or (1, p). 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. For example, s = 1 − 3 = −2 for (+, −, −, −) and its mirroring s' = −s = +2 for (−, +, +, +).

The signature of a metric tensor is defined as the signature of the corresponding quadratic form. 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. Usually, is required, which is the same as saying a metric tensor must be nondegenerate, i. e. no nonzero vector is orthogonal to all vectors. By Sylvester's law of inertia, the numbers (v, p, r) are basis independent.

By the spectral theorem a symmetric n × n matrix over the reals is always diagonalizable, and has therefore exactly n real eigenvalues (counted with algebraic multiplicity). Thus 1=v + p = n = dim(V).

According to Sylvester's law of inertia, the signature of the scalar product (a. k. a. real symmetric bilinear form), g does not depend on the choice of basis. Moreover, for every metric g of signature (v, p, r) there exists a basis such that
1=gab = +1 for 1=a = b = 1, , v, 1=gab = −1 for 1=a = b = v + 1, , v + p and 1=gab = 0 otherwise. It follows that there exists an isometry (V1, g1) → (V2, g2) if and only if the signatures of g1 and g2 are equal. 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.

The number v (resp. p) is the maximal dimension of a vector subspace on which the scalar product g is positive definite (resp. negative definite), and r is the dimension of the radical of the scalar product g or the null subspace of symmetric matrix gab of the scalar product. Thus a nondegenerate scalar product has signature (v, p, 0), with 1=v + p = n. A duality of the special cases (v, p, 0) correspond to two scalar eigenvalues which can be transformed into each other by the mirroring reciprocally.

There are some methods for computing the signature of a matrix. For any nondegenerate symmetric n × n matrix, diagonalize it (or find all of eigenvalues of it) and count the number of positive and negative signs. For a symmetric matrix, the characteristic polynomial will have all real roots whose signs may in some cases be completely determined by Descartes' rule of signs. Lagrange's algorithm gives a way to compute an orthogonal basis, and thus compute a diagonal matrix congruent (thus, with the same signature) to the other one: the signature of a diagonal matrix is the number of positive, negative and zero elements on its diagonal. According to Jacobi's criterion, a symmetric matrix is positive definite if and only if all the determinants of its main minors are positive.

If a metric is regular everywhere then the signature of the metric is constant. However if one allows for metrics that are degenerate or discontinuous on some hypersurfaces, then signature of the metric may change at these surfaces. Such signature changing metrics may possibly have applications in cosmology and quantum gravity.