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:
![{\displaystyle M\;[\mathbf {s} \;\mathbf {t} ]\;=\;[\mathbf {s} \;\mathbf {t} ]\Lambda _{\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c1740b54fe87a89a53b88b5e5d4823ee18d74d3)
where s and t are four-dimensional column vectors and Λθ 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.
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