Editing Metric signature (section)

{{Short description|Number of positive, negative and zero eigenvalues of a metric tensor}}
In [[mathematics]], the '''signature''' of a [[metric tensor]] ''g'' (or equivalently, a [[Real number|real]] [[quadratic form]] thought of as a real [[symmetric bilinear form]] on a [[Dimension (vector space)|finite-dimensional]] [[vector space]]) is the number (counted with multiplicity) of positive, negative and zero [[eigenvalue]]s of the real [[symmetric matrix]] {{nowrap|''g''<sub>''ab''</sub>}}  of the metric tensor with respect to a [[Basis (linear algebra)|basis]]. In [[Relativity (physics)|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 [[Linear subspace|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. It is denoted by three [[integer]]s {{nowrap|(''v'', ''p'', ''r'')}}, where v is the number of positive eigenvalues, p is the number of negative ones and r is the number of zero eigenvalues of the metric tensor. It can also be denoted {{nowrap|(''v'', ''p'')}} implying ''r''&thinsp;=&thinsp;0, or as an explicit list of signs of eigenvalues such as {{nowrap|(+, −, −, −)}} or {{nowrap|(−, +, +, +)}} for the signatures {{nowrap|(1, 3, 0)}} and {{nowrap|(3, 1, 0)}}, respectively.<ref>Rowland, Todd. "Matrix Signature." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/MatrixSignature.html</ref>

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 [[definite bilinear form|positive definite]] signature {{nowrap|(''v'', 0)}}. A [[Lorentz metric|Lorentzian metric]] is a metric with signature {{nowrap|(''p'', 1)}}, or {{nowrap|(1, ''p'')}}.

There is another notion of '''signature''' of a nondegenerate metric tensor given by a single number ''s'' defined as {{nowrap|(''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&thinsp;− 3 = −2 for {{nowrap|(+, −, −, −)}} and its mirroring ''s' '' = −''s'' = +2 for {{nowrap|(−, +, +, +)}}.