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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 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 *S ^{3}*) represented within four-dimensional Euclidean space:

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.

## See also

## External links

- possible relevance of eigenplanes in cosmology
- GNU GPL software for calculating eigenplanes
- Proof constructed by J M Shelley 2017