|
|
| Line 1: |
Line 1: |
| In [[mathematics]], an '''eigenplane''' is a two-[[dimension]]al [[invariant subspace]] in a given [[vector space]]. By analogy with the term ''[[eigenvector]]'' for a vector which, when operated on by a [[linear transformation|linear operator]] is another vector which is a [[scalar (mathematics)|scalar]] multiple of itself, the term '''''eigenplane''''' can be used to describe a two-dimensional [[plane (mathematics)|plane]] (a ''2-plane''), such that the operation of a [[linear transformation|linear operator]] on a vector in the 2-plane always yields another vector in the same 2-plane.
| | Bricklayer Dee Gohr from Aberdeen, loves ghost hunting, diet and architecture. Intends to give up work and take the family to a lot of the great heritage listed locales on the globe like Chongoni Rock-Art Area. |
| | |
| A particular case that has been studied is that in which the linear operator is an [[isometry]] ''M'' of the [[hypersphere]] (written ''S<sup>3</sup>'') represented within four-dimensional [[Euclidean space]]:
| |
| | |
| :<math>M \; [ \mathbf{s} \; \mathbf{t} ] \; = \; [ \mathbf{s} \; \mathbf{t} ] \Lambda_\theta </math>
| |
| | |
| where '''s''' and '''t''' are four-dimensional column vectors and Λ<sub>θ</sub> is a two-dimensional '''eigenrotation''' within the '''eigenplane'''.
| |
| | |
| In the usual [[eigenvector]] problem, there is freedom to multiply an eigenvector by an arbitrary scalar; in this case there is freedom to multiply by an arbitrary non-zero [[rotation]].
| |
| | |
| This case is potentially physically interesting in the case that the [[shape of the universe]] is a [[multiply connected]] [[3-manifold]], since finding the [[angle]]s of the eigenrotations of a candidate isometry for [[shape of the universe|topological lensing]] is a way to [[scientific method|falsify]] such hypotheses.
| |
| | |
| ==See also==
| |
| * [[Bivector]]
| |
| * [[Plane of rotation]]
| |
| | |
| ==External links==
| |
| *[http://arxiv.org/abs/astro-ph/0409694 possible relevance of eigenplanes] in [[physical cosmology|cosmology]]
| |
| *[http://cosmo.torun.pl/GPLdownload/eigen/ GNU GPL software for calculating eigenplanes]
| |
| [[Category:Linear algebra]]
| |
Latest revision as of 22:09, 17 June 2014
Bricklayer Dee Gohr from Aberdeen, loves ghost hunting, diet and architecture. Intends to give up work and take the family to a lot of the great heritage listed locales on the globe like Chongoni Rock-Art Area.