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In [[mathematics]], in [[linear algebra]], the '''cyclic decomposition theorem''' is an assertion of a certain property of [[dimension|finite-dimenstional]] [[vector space]]s in relation to [[linear transformation]]s of the spaces. The theorem states that given a linear transformation of a finite-dimensional vector space over an [[algebraically closed field]], the vector space can be expressed as a [[direct sum]] of [[linear subspace|subspace]]s each of which is [[Invariant subspace|invariant]] under the transformation and is cyclically generated by the transformation. This result is considered to be "one of the deepest results in linear algebra".<ref name=LA>{{cite book|last=Kenneth Hoffmann, Ray Kunze|title=Linear Algebra (Second Edition)|year=1971|publisher=Prentice-Hall|page=231}}</ref>
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==Preliminaries==
A knowledge of certain concepts and terminology related to linear transformations is an essential prerequisite for stating and understanding the cyclic decomposition theorem and its proof. To explain these, let <math>T</math> be a [[linear operator]] on a finite-dimensional vector space <math>V</math> over a [[Field (mathematics)|field]] <math>F</math>. Let <math>v</math> be a vector in <math>V</math>.  
 
===Cyclic subspace===
 
The subspace of <math>V</math> generated by the set <math>\{ v, Tv, T^2v, \ldots, T^kv, \ldots\}</math> is called the <math>T</math>-''[[cyclic subspace]] generated by'' <math>v</math>. This subspace is denoted by <math>Z(v;T)</math>.
 
===Annihilator of a vector===
 
Let <math>F[x]</math> be the [[Ring (mathematics)|ring]] of all [[polynomial]]s in <math>x</math> over the field <math>F</math>.
The set of all polynomials <math>g(x)</math> in <math>F[x]</math> such that <math>g(T)v=0</math> is called the <math>T</math>-''annihilator'' of <math>v</math>. It is denoted by <math>M(v;T)</math>. <math>M(v;T)</math> is an ideal in the ring <math>F[x]</math>. The ideal <math>M(v;T)</math> consists of all multiples by elements of <math>F[x]</math> of some fixed monic polynomial in <math>F[x]</math>. This fixed  monic polynomial is denoted by <math>p_v(x)</math> and it is also sometimes referred to as the <math>T</math>-annihilator of <math>v</math>.
 
===Conductor===
 
Let <math>W</math> be subspace of <math>V</math> which is invariant under <math>T</math>. Let <math>F[x]</math> be the [[Ring (mathematics)|ring]] of all [[polynomial]]s in <math>x</math> over the field <math>F</math>.
The set of all polynomials <math>g(x)</math> in <math>F[x]</math> such that <math>g(T)v\in W</math> is called the <math>T</math>-''conductor of <math>v</math> into <math>W</math>'' and is denoted by <math>S_T(v;W)</math>. <math>S_T(v;W)</math> is an ideal in the ring <math>F[x]</math>.  The unique monic polynomial <math>g(x)</math> of least degree such that <math>g(T)v\in W</math>, which is the generator of the ideal <math>S_T(v;W)</math>, is also called the <math>T</math>-''conductor of <math>v</math> into <math>W</math>''.
 
===Admissible subspace===
 
Let <math>W</math> be a [[linear subspace]] of <math>V</math>. <math>W</math> is called a <math>T</math>-admissible subspace of <math>V</math> if the following conditions are satisfied.
# <math>W</math> is invariant under <math>T</math>;  that is, for any <math>w</math> in <math>W</math>, the vector <math>Tw</math> is in <math>W</math>.  
# For any polynomial <math>f(x)</math> in <math>F[x]</math> and any vector <math>v</math> in <math>V</math>, if <math>f(T)v</math> is in <math>W</math> then there is a vector <math>w</math> in <math>W</math> such that <math>f(T)v=f(T)w</math>.
 
==Cyclic decomposition theorem==
Let <math>T</math>  be a linear operator on a finite-dimensional vector space <math>V</math> and let <math>W_0</math> be a proper <math>T</math>-admissible subspace of <math>V</math>. There exist non-zero vectors <math>v_1,v_2, \ldots, v_r</math> in <math>V</math> with respective <math>T</math>-annihilators <math>p_1(x),p_2(x), \ldots, p_r(x)</math> such that
 
# <math> V = W_0 \oplus Z(v_1;T) \oplus Z(v_2;T) \oplus \cdots \oplus Z(v_r;T)</math>.  
# <math>p_k(x)</math>  divides <math>p_{k-1}(x)</math> for <math> k = 2, \ldots , r</math>.
 
Furthermore, the integer <math>r</math> and the annihilators <math>p_1(x),p_2(x), \ldots, p_r(x)</math> are uniquely
determined by (1), (2), and the fact that no <math> v_k</math> is 0.
 
==References==
{{reflist}}
 
==External links==
 
*[http://planetmath.org/cyclicdecompositiontheorem PlanetMath: Cyclic decomposition theorem]
 
 
[[Category:Linear algebra]]
[[Category:Vector spaces]]

Latest revision as of 08:22, 6 October 2014

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