cyclic decomposition theorem

Let k be a field, V a finite dimensional vector spaceMathworldPlanetmath over k and T a linear operator over V. Call a subspacePlanetmathPlanetmathPlanetmath WV T-admissible if W is T-invariant and for any polynomialPlanetmathPlanetmath f(X)k[X] with f(T)(v)W for vV, there is a wW such that f(T)(v)=f(T)(w).

Let W0 be a proper T-admissible subspace of V. There are non zero vectors x1,,xr in V with respective annihilator polynomials p1,,pr such that

  1. 1.

    V=W0Z(x1,T)Z(xr,T) (See the cyclic subspace definition)

  2. 2.

    pk divides pk-1 for every k=2,,r

Moreover, the integer r and the minimal polynomialsPlanetmathPlanetmath ( p1,,pr are uniquely determined by (1),(2) and the fact that none of xk is zero.

This is “one of the deepest results in linear algebraMathworldPlanetmath” (Hoffman & Kunze)

Title cyclic decomposition theorem
Canonical name CyclicDecompositionTheorem
Date of creation 2013-03-22 14:05:10
Last modified on 2013-03-22 14:05:10
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Theorem
Classification msc 15A04
Synonym T-admissible
Synonym T-admissible
Related topic CyclicSubspace
Defines admissible subspace