One method for performing orthogonalization is the Gram-Schmidt process.
When performing orthogonalization on a computer, the Householder transformation is usually preferred over the Gram-Schmidt process since it is more numerically stable, i.e. rounding errors tend to have less serious effects.
In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.
For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.
Feeds |
Contact