signature -- outputs the signature of a symmetric bilinear form over the real or rational numbers

Synopsis

• Usage:

  signature(beta)

• Inputs:
  • beta, a Grothendieck Witt Class, a symmetric bilinear form defined over $\mathbb{Q}$ or $\mathbb{R}$
• Outputs:
  • n, an integer, the signature of the symmetric bilinear form $\beta$

Description

Given a symmetric bilinear form, after diagonalizing it, we can consider the number of positive entries minus the number of negative entries appearing along the diagonal. This is the signature of a symmetric bilinear form, and is one of the primary invariants we use to classify forms. For more information see gwIsomorphic.

i1 : M = matrix(RR,{{0,0,1},{0,1,0},{1,0,0}});

                3         3
o1 : Matrix RR    <-- RR
              53        53

i2 : beta = gwClass(M);

i3 : signature(beta)

o3 = 1

See also

• gwIsomorphic -- determines whether two Grothendieck Witt classes are isomorphic over CC, RR, QQ, or a finite field.
• sumDecomposition -- produces a simplified diagonal representative of a Grothendieck Witt class
• sumDecompositionString -- produces a simplified diagonal representative of a Grothendieck Witt class