Signature (mathematics) - Definition

In mathematics, signature can refer to

• The signature of a permutation is ±1 according to whether it is an even/odd permutation. The signature function defines a group homomorphism from the symmetric group to the group {±1}.

• The signature of a real quadratic form (or a symmetric bilinear form) is the number of positive, negative, and zero eigenvalues of the corresponding matrix. The signature is an invariant of the quadratic form (i.e. independent of the choice of basis).
  • Related to this is the metric signature of the metric tensor on a pseudo-Riemannian manifold.
  • Also related is the topological signature of a 4k-dimensional compact, orientable manifold. One can define a symmetric bilinear form on the middle de Rham cohomology group H^2k(M). The topological signature of M is the signature of this form. See also: signature complex.

• Signature (universal algebra)