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. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} In mathematics the term diagonalization may refer to A Diagonalizable matrix, which can be put into a form with nonzero entries only on the main diagonal 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 (−,+,+,+).
The signature is said to be indefinite or mixed if both p and q are non-zero. A Riemannian metric is a metric with a (positive) definite signature. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In mathematics positive definite may refer to Positive-definite matrix Positive-definite function Positive definite A Lorentzian metric is one with signature (p, 1) (or sometimes (1, q)). In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian 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. In Mathematics, codimension is a basic geometric idea that applies to Subspaces in Vector spaces and more generally to Submanifolds in Manifolds In Linear algebra, a positive-definite matrix is a (Hermitian matrix which in many ways is analogous to a Positive Real number. Using the nondegenerate metric tensor from above, the signature is simply the minimum of p and q. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space For example (+,−,−,−) and (−,+,+,+) have both signature s = 1.
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Definition
Let A be a symmetric matrix of reals. In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} The metric signature (i + ,i − ,i0) of A is a group of three natural numbers defined as the number of positive, negative and zero-valued eigenvalues of the matrix counted with regard to their algebraic multiplicity. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes
If φ is a scalar product on a finite dimension vector space V, the signature of the matrix which represents φ for a basis. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the dimension of a Vector space V is the cardinality (i In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Basis vector redirects here For basis vector in the context of crystals see Crystal structure. According to Sylvester's law of inertia the signature does not depend on the basis. In Linear algebra, Sylvester's law of inertia is a Theorem describing a canonical representative for a real symmetric matrix under Congruence
Properties
Spectral theorem
Due to the spectral theorem a symmetric matrix of reals is always diagonalizable. In Mathematics, particularly Linear algebra and Functional analysis, the spectral theorem is any of a number of results about Linear operators In Linear algebra, a Square matrix A is called diagonalizable if it is similar to a Diagonal matrix, i Moreover, it has exactly n eigenvalues (counted according by their algebraic multiplicity). In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Thus i + + i − + i0 = n
Sylvester's law of inertia
According to Sylvester's law of inertia two scalar products are isometrical if and only if they have the same signature. In Linear algebra, Sylvester's law of inertia is a Theorem describing a canonical representative for a real symmetric matrix under Congruence This means that the signature is a complete invariante for scalar products on isometric transformations. In the same way two symmetric matrices are congruent if and only if they have the same signature. See Congruence (geometry for the term as used in elementary geometry
Geometrical interpretation of the indices
The values i + ,i − e i0 are called add name, add name and add name. The i0 is the dimension of the radical of the scalar product φ and the null space of A. A symmetric bilinear form is as the name implies a Bilinear form on a Vector space that is symmetric Thus a non degenerate scalar product has signature (i + ,i − ,0).
The indices i + e i − are the greatest dimension of a vector subspace on which the scalar product is positive-definite and negative-definite. The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics.
Examples
Matrices
The signature of the identity matrix 
is (n,0,0). In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main More generally, the signature of a diagonal matrix is the number of positive, negative and zero numbers on its main diagonal. In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Linear algebra, the main diagonal (sometimes leading diagonal or primary diagonal) of a matrix A is the collection of cells A_{ij}
The following matrices have both the same signature (1,1,0), therefore they are congruent because of Sylvester's law of inertia:
Scalar products
The standard scalar product defined on 
has (n,0,0) signature. A scalar product has this signature if and only if it is a positive definite scalar product. In mathematics positive definite may refer to Positive-definite matrix Positive-definite function Positive definite
A negative defined scalar product has (0,n,0) signature. A semi-definite positive scalar product has (n,0,m) signature.
The Minkowski space is 
and has a scalar product defined by the matrix
and has signature (1,3,0). In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity Sometimes it is used with the opposite signs, thus obtaining (3,1,0) signature.
How to compute the signature
There are some methods for computing the signature of a matrix.
- The sign of the roots of the characteristic polynomial may be determined by Cartesius' sign rule as long as all roots are reals.
- Lagrange algorithm avails 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. In Mathematics, an orthonormal basis of an Inner product space V (i In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero
- According to Jacobi's criterion, a symmetric matrix is positive-definite if and only if all the determinants of its main minors are positive.
Signature in physics
In theoretical physics, spacetime is modeled by a pseudo-Riemannian manifold. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. The signature counts how many dimensions of spacetime have a time-like or space-like character, in the sense defined by special relativity. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial
The spacetimes with purely space-like directions are said to have Euclidean signature while the spacetimes with signature like (3,1) are said to have Minkowskian signature. The more general signatures are often referred to as Lorentzian signature although this term is often used as a synonym of the Minkowskian signature.
See also pseudo-Riemannian manifold. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.
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