Editing Metric signature (section)

== Properties ==

=== Signature and dimension ===
By the [[spectral theorem]] a symmetric {{nowrap|''n''&thinsp;×&thinsp;''n''}} matrix over the reals is always [[diagonalizable]], and has therefore exactly ''n'' real eigenvalues (counted with [[algebraic multiplicity]]). Thus {{nowrap|1=''v'' + ''p'' = ''n'' = dim(''V'')}}.

=== Sylvester's law of inertia: independence of basis choice and existence of orthonormal basis ===
According to [[Sylvester's law of inertia]], the signature of the scalar product  (a.k.a. real symmetric bilinear form), ''g'' does not depend on the choice of basis. Moreover, for every metric ''g'' of signature {{nowrap|(''v'', ''p'', ''r'')}} there exists a basis such that 
{{nowrap|1=''g''<sub>''ab''</sub> = +1}} for {{nowrap|1=''a'' = ''b'' = 1, ..., ''v''}}, {{nowrap|1=''g''<sub>''ab''</sub> = −1}} for {{nowrap|1=''a'' = ''b'' = ''v'' + 1, ..., ''v'' + ''p''}} and {{nowrap|1=''g''<sub>''ab''</sub> = 0}} otherwise. It follows that there exists an [[isometry]] {{nowrap|(''V''<sub>1</sub>, ''g''<sub>1</sub>) → (''V''<sub>2</sub>, ''g''<sub>2</sub>)}} if and only if the signatures of ''g''<sub>1</sub> and ''g''<sub>2</sub> are equal.  Likewise the signature is equal for two [[congruent matrices]] and classifies a matrix up to congruency. Equivalently, the signature is constant on the orbits of the [[general linear group]] GL(''V'') on the space of symmetric rank 2 contravariant tensors ''S''<sup>2</sup>''V''<sup>∗</sup> and classifies each orbit.

===Geometrical interpretation of the indices ===
The number ''v'' (resp.  ''p'') is the maximal dimension of a vector subspace on which the scalar product ''g'' is positive-definite (resp. negative-definite), and ''r'' is the dimension of the [[Symmetric bilinear form#Orthogonality and singularity|radical]] of the scalar product ''g'' or the [[null space|null subspace]] of [[symmetric matrix]] {{nowrap|''g''<sub>''ab''</sub>}} of the [[scalar product]]. Thus a nondegenerate scalar product has signature {{nowrap|(''v'', ''p'', 0)}}, with {{nowrap|1=''v'' + ''p'' = ''n''}}. A duality of the special cases {{nowrap|(''v'', ''p'', 0)}} correspond to two scalar eigenvalues which can be transformed into each other by the mirroring reciprocally.