canonical basis for symmetric bilinear forms

If B:V×VK is a symmetric bilinear formMathworldPlanetmath over a finite-dimensional vector spaceMathworldPlanetmath, where the characteristic of the field is not 2, then we may prove that there is an orthogonal basis such that B is represented by


Recall that a bilinear formPlanetmathPlanetmath has a well-defined rank, and denote this by r.

If K= we may choose a basis such that a1==at=1, at+1==at+p=-1 and at+p+j=0, for some integers p and t, where 1jn-t-p. Furthermore, these integers are invariants of the bilinear form. This is known as Sylvester’s Law of Inertia. B is positive definitePlanetmathPlanetmath if and only if t=n, p=0. Such a form constitutes a real inner product spaceMathworldPlanetmath.

If K= we may go further and choose a basis such that a1==ar=1 and ar+j=0, where 1jn-r.

If K=Fp we may choose a basis such that a1==ar-1=1,

ar=n or ar=1; and ar+j=0, where 1jn-r, and n is the least positivePlanetmathPlanetmath quadratic non-residue.

Title canonical basis for symmetric bilinear forms
Canonical name CanonicalBasisForSymmetricBilinearForms
Date of creation 2013-03-22 14:56:25
Last modified on 2013-03-22 14:56:25
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 7
Author Mathprof (13753)
Entry type Definition
Classification msc 47A07
Classification msc 11E39
Classification msc 15A63
Defines Sylvester’s Law of Inertia