Eigenplane

In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. By analogy with the term "eigenvector" for a vector which, when operated on by a linear operator is another vector which is a scalar multiple of itself, the term "eigenplane" can be used to describe a two-dimensional plane (a "2-plane"), such that the operation of a linear operator on a vector in the 2-plane always yields another vector in the same 2-plane.

A particular case that has been studied is that in which the linear operator is an isometry "M" of the hypersphere (written "S³") represented within four-dimensional Euclidean space:

:$M\; ;\; [\; mathbf\{s\}\; ;\; mathbf\{t\}\; ]\; ;\; =\; ;\; [\; mathbf\{s\}\; ;\; mathbf\{t\}\; ]\; Lambda\_\; heta$

where s and t are four-dimensional column vectors and Λ_θ 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 angles of the eigenrotations of a candidate isometry for topological lensing is a way to falsify such hypotheses.

External links

* [http://arxiv.org/abs/astro-ph/0409694 possible relevance of eigenplanes] in cosmology
* [http://cosmo.torun.pl/GPLdownload/eigen/ GNU GPL software for calculating eigenplanes]