Metric signature

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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 (−,+,+,+).

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)).

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 nondegenerate metric tensor from above, the signature is simply the minimum of p and q. For example (+,−,−,−) and (−,+,+,+) have both signature s = 1.

Contents 1 Definition 2 Properties 2.1 Spectral theorem 2.2 Sylvester's law of inertia 2.3 Geometrical interpretation of the indices 3 Examples 3.1 Matrices 3.2 Scalar products 4 How to compute the signature 5 Signature in physics

[edit] Definition

Let A be a symmetric matrix of reals. 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.

If φ is a scalar product on a finite dimension vector space V, the signature of the matrix which represents φ for a basis. According to Sylvester's law of inertia the signature does not depend on the basis.

[edit] Properties

[edit] Spectral theorem

Due to the spectral theorem a symmetric matrix of reals is always diagonalizable. Moreover, it has exactly n eigenvalues (counted according by their algebraic multiplicity). Thus i ₊ + i ₋ + i₀ = n

[edit] 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. 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.

[edit] 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. 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.

[edit] Examples

[edit] Matrices

The signature of the identity matrix is (n,0,0). More generally, the signature of a diagonal matrix is the number of positive, negative and zero numbers on its main diagonal.

The following matrices have both the same signature (1,1,0), therefore they are congruent because of Sylvester's law of inertia:

[edit] 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.

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

ed ha quindi segnatura (1,3,0). Sometimes it is used with the opposite signs, thus obtaining (3,1,0) signature.

[edit] How to compute the signature

There are some methods for computing the signature of a matrix.

• The sign of the roots of the caracteristic 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.
• According to Jacobi's criterion, a symmetric matrix is positive-definite if and only if all the determinants of its main minors are positive.

[edit] Signature in physics

In theoretical physics, spacetime is modeled by a pseudo-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.

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 synonyn of the Minkowskian signature.

See also pseudo-Riemannian manifold.