Existence and construction of two-dimensional invariant subspaces for pairs of rotations

Ernst Dieterich¹

¹ Uppsala universitet Box 480 SE-751 06 Uppsala, Sweden

Colloquium Mathematicum, Tome 114 (2009) no. 2, pp. 203-211

Résumé

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 $\alpha \in [0,\pi ]$. 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.

Détail

DOI : 10.4064/cm114-2-4

Keywords: rotation euclidean space even dimension mean orthogonal linear operator which orthogonal direct sum rotations dimensional linear subspaces common angle alpha present criterion existence dimensional subspace which invariant under given pair rotations terms vanishing determinant associated pair criterion constructive whenever satisfied prove every pair rotations has dimensional invariant subspace only dimension congruent modulo