primary decomposition theorem

This is an important theorem in linear algebraMathworldPlanetmath. It states the following: Let k be a field, V a vector spaceMathworldPlanetmath over k, dimV=n, and T:VV a linear operator, such that its minimal polynomial (or its annihilator polynomial) is mT, which decomposes in k[X] into irreducible factors as mT=p1α1prαr. Then,

  1. 1.


  2. 2.

    ker(piαi(T)) is T-invariant for every i

  3. 3.

    If Ti is the restrictionPlanetmathPlanetmath of T to ker(piαi(T)), then mTi=piαi

This is a consequence of a more general theorem: Let V, T be as above, and fk[X] such that f(T)=0, with f=f1fr and (fi,fj)=1 if ij, then

  1. 1.


  2. 2.

    ker(fi(T)) is T-invariant for every i

To illustrate its importance, the primary decomposition theoremPlanetmathPlanetmath, together with the cyclic decomposition theorem, imply the existence and uniqueness of the Jordan canonical formMathworldPlanetmath.

Title primary decomposition theorem
Canonical name PrimaryDecompositionTheorem
Date of creation 2013-03-22 14:15:30
Last modified on 2013-03-22 14:15:30
Owner gumau (3545)
Last modified by gumau (3545)
Numerical id 8
Author gumau (3545)
Entry type Theorem
Classification msc 15A04