By a rotation in a Euclidean space V of even dimension we mean an orthogonal linear operator on V which is an orthogonal direct sum of rotations in 2-dimensional linear subspaces of V by a common angle α ∈ [0,π]. We present a criterion for the existence of a 2-dimensional subspace of V which is invariant under a given pair of rotations, in terms of the vanishing of a determinant associated with that pair. This criterion is constructive, whenever it is satisfied. It is also used to prove that every pair of rotations in V has a 2-dimensional invariant subspace if and only if the dimension of V is congruent to 2 modulo 4.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-4,
author = {Ernst Dieterich},
title = {Existence and construction of two-dimensional invariant subspaces for pairs of rotations},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {203-211},
zbl = {1161.15001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-4}
}
Ernst Dieterich. Existence and construction of two-dimensional invariant subspaces for pairs of rotations. Colloquium Mathematicae, Tome 116 (2009) pp. 203-211. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-4/