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Eigenplane

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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 transformation linear operator is another vector which is a 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. 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_\theta

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 shape of the universe topological lensing is a way to scientific method falsify such hypotheses.

See also
*eigenvector

External links

- possible relevance of eigenplanes in cosmology
- GNU GPL software for calculating eigenplanes Category:Linear algebra fr:plan propre