canonical basis for symmetric bilinear forms
If is a symmetric bilinear form
over a finite-dimensional vector space
, where the characteristic of the field is
not 2,
then we may prove that there is an orthogonal basis such that is represented by
Recall that a bilinear form has a well-defined rank, and denote this by .
If we may choose a basis such that ,
and , for some integers and ,
where .
Furthermore, these integers are invariants of the bilinear form.
This is known as Sylvester’s Law of Inertia.
is positive definite if and only if
, . Such a form constitutes a real inner product space
.
If we may go further and choose a basis such that and , where .
If we may choose a basis such that ,
or ;
and , where , and
is the least positive 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 |