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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.
Ernst Dieterich. "Existence and construction of two-dimensional invariant subspaces for pairs of rotations." Colloquium Mathematicae 114.2 (2009): 203-211. <http://eudml.org/doc/284228>.
@article{ErnstDieterich2009, abstract = {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.}, author = {Ernst Dieterich}, journal = {Colloquium Mathematicae}, keywords = {pair of rotations; invariant subspace; commutator determinant; skew-symmetric matrix; division algebras; Euclidean space; orthogonal linear operator}, language = {eng}, number = {2}, pages = {203-211}, title = {Existence and construction of two-dimensional invariant subspaces for pairs of rotations}, url = {http://eudml.org/doc/284228}, volume = {114}, year = {2009}, }
TY - JOUR AU - Ernst Dieterich TI - Existence and construction of two-dimensional invariant subspaces for pairs of rotations JO - Colloquium Mathematicae PY - 2009 VL - 114 IS - 2 SP - 203 EP - 211 AB - 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. LA - eng KW - pair of rotations; invariant subspace; commutator determinant; skew-symmetric matrix; division algebras; Euclidean space; orthogonal linear operator UR - http://eudml.org/doc/284228 ER -