Signature (mathematics)

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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 symmetric bilinear form (or a real quadratic form) is the number of positive, negative, and zero eigenvalues of the corresponding matrix. The signature is an invariant of the quadratic form by Sylvester's law of inertia (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, oriented manifold. The cup product defines a symmetric 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.
• The signature of a flag of subspaces is the sequence of the dimensions of the subspaces.
• In logic, the signature of a structure contains information about the names and arities of its functions and relations.
• Signature of a knot