The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted. If the matrix is n×n, the possible number of positive signs may take any value p from 0 to n. The signature may be denoted either by a pair of integers such as (p, q), or as an explicit list such as (−,+,+,+). In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... The term diagonalization is used in two different senses in mathematics: The process of finding a diagonal matrix similar to a given square matrix or representing a given linear map. ...

The signature is said to be indefinite if both p and q are non-zero. A Riemannian metric is a metric with a (positive) definite signature. A Lorentzian metric is one with signature (p, 1) (or sometimes (1, q)). In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

There is also another definition of signature which uses a single number s defined as the codimension of the biggest (positive or negative) definite subspace. Using the nondegemerate metric tensor from above, the signature is simply the minimum of p and q. For example (+,−,−,−) and (−,+,+,+) have both signature s = 1. In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and more generally to submanifolds in manifolds, and suitable subsets of algebraic varieties. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real numbers. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

See also pseudo-Riemannian manifold. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...