cyclic subspace

Let V be a vector spaceMathworldPlanetmath over a field k, and xV. Let T:VV be a linear transformation. The T-cyclic subspace generated by x is the smallest T-invariant subspacePlanetmathPlanetmath which contains x, and is denoted by Z(x,T).

Since x,T(x),,Tn(x),Z(x,T), we have that


On the other hand, since W is T-invariant, Z(x,T)W. Hence Z(x,T) is the subspacePlanetmathPlanetmathPlanetmath generated by x,T(x),,Tn(x), In other words, Z(x,T)={p(T)(x)pk[X]}.

Remark. If Z(x,T)=V we say that x is a cyclic vector of T.

Title cyclic subspace
Canonical name CyclicSubspace
Date of creation 2013-03-22 14:05:03
Last modified on 2013-03-22 14:05:03
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 12
Author CWoo (3771)
Entry type Definition
Classification msc 15A04
Classification msc 47A16
Synonym cyclic vector subspace
Related topic CyclicDecompositionTheorem
Related topic CyclicVectorTheorem
Defines cyclic vector