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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 of the metric tensor is n×n, then the number of positive and negative eigenvalues p and q = np may take a pair of values from 0 to n. The signature may be denoted either by a pair of integers such as (p, −q) or (q, −p), or as an explicit list such as (+, −, −, −) or (−, +, +, +) , in this case (1,3) resp. (3,1).[1]

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. 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 number pq, where the p and q are the number of positive and negative eigenvalues of the metric tensor. Using the nondegenerate metric tensor from above, the signature is simply the sum of p and - q. For example, s = (1 − 3) = −2 for (+,−,−,−) and s = (3 − 1) = +2 for (−, +, +, +).

Contents

Definition

Let A be a symmetric matrix of reals. More generally, the metric signature (i+,i,i0) of A is a group of three natural numbers can be defined as the number of positive, negative and zero-valued eigenvalues of the matrix counted with regard to their algebraic multiplicity. In the case i0 is non-zero, the matrix A called degenerate.

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

Properties

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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

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.

Geometrical interpretation of the indices

The indices i + and i are the dimensions of the two vector subspaces on which the scalar product is positive-definite and negative-definite respectively. And the i0 is the dimension of the radical of the scalar product φ or the null subspace of symmetric matrix A of the bilinear form. Thus a non degenerate scalar product has signature (i + ,i ,0), with i = ni + . So the values i + ,i and i0 are also called the dimensions of the positive-definite, negative-definite and null vector subspaces of the whole vector space V which correspond to the matrix A. The special cases (n,0,0) and (0,n,0) correspond to the two equivalent vector spaces on which the scalar product is positive-definite and negative-definite respectively, and can transform each other by multiplying -1 to their scalar product.

Examples

Matrices

The signature of the identity matrix n\times n 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:

\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.

Scalar products

The standard scalar product defined on  \mathbb{R}^n has (n,0,0) signature. A scalar product has this signature if and only if it is a positive definite scalar product.

A negative definite scalar product has (0,n,0) signature. A semi-definite positive scalar product has (n,0,m) signature.

The Minkowski space is \R^4 and has a scalar product defined by the matrix

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}

and has signature (1,3,0). 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.

  • For any nondegenerate symmetric matrix of n×n, diagonalize it (or find all of eigenvalues of it) and count the number of positive and negative signs, and get p and q = np, they may take a pair of values from 0 to n, then the signature will be s = pq.
  • 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.
  • 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. The signature counts how many time-like or space-like characters in the spacetime, 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 synonym of the Minkowskian signature.

See also

Notes

  1. ^ Rowland, Todd. "Matrix Signature." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/MatrixSignature.html

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