Characteristic invariant subspaces generated by a single vector. (English) Zbl 1309.15017
Summary: If \(f\) is an endomorphism of a finite dimensional vector space \(V\) over a field \(K\) then an invariant subspace \(X \subseteq V\) is called hyperinvariant (respectively, characteristic) if \(X\) is invariant under all endomorphisms (respectively, automorphisms) that commute with \(f\). The characteristic hull of a subset \(W\) of \(V\) is defined to be the smallest characteristic subspace in \(V\) that contains \(W\). It is known that characteristic subspaces that are not hyperinvariant can only exist when \(| K | = 2\). In this paper we study subspaces \(X\) which are the characteristic hull of a single element. In the case where \(| K | = 2\) we derive a necessary and sufficient condition such that \(X\) is hyperinvariant.
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 47A15 | Invariant subspaces of linear operators |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
Keywords:
characteristic subspaces; hyperinvariant subspaces; invariant subspaces; elementary divisors; characteristic hull; exponent; height; generatorsReferences:
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