primary decomposition theorem
This is an important theorem in linear algebra^{}. It states the following: Let $k$ be a field, $V$ a vector space^{} over $k$, $dimV=n$, and $T:V\to V$ a linear operator, such that its minimal polynomial (or its annihilator polynomial) is ${m}_{T}$, which decomposes in $k[X]$ into irreducible factors as ${m}_{T}={p}_{1}^{{\alpha}_{1}}\mathrm{\dots}{p}_{r}^{{\alpha}_{r}}$. Then,

1.
$V={\oplus}_{i=1}^{r}\mathrm{ker}({p}_{i}^{{\alpha}_{i}}(T))$

2.
$\mathrm{ker}({p}_{i}^{{\alpha}_{i}}(T))$ is $T$invariant for every $i$

3.
If ${T}_{i}$ is the restriction^{} of $T$ to $\mathrm{ker}({p}_{i}^{{\alpha}_{i}}(T))$, then ${m}_{{T}_{i}}={p}_{i}^{{\alpha}_{i}}$
This is a consequence of a more general theorem: Let $V$, $T$ be as above, and $f\in k[X]$ such that $f(T)=0$, with $f={f}_{1}\mathrm{\dots}{f}_{r}$ and $({f}_{i},{f}_{j})=1$ if $i\ne j$, then

1.
$V={\oplus}_{i=1}^{r}\mathrm{ker}({f}_{i}(T))$

2.
$\mathrm{ker}({f}_{i}(T))$ is $T$invariant for every $i$
To illustrate its importance, the primary decomposition theorem^{}, together with the cyclic decomposition theorem, imply the existence and uniqueness of the Jordan canonical form^{}.
Title  primary decomposition theorem 

Canonical name  PrimaryDecompositionTheorem 
Date of creation  20130322 14:15:30 
Last modified on  20130322 14:15:30 
Owner  gumau (3545) 
Last modified by  gumau (3545) 
Numerical id  8 
Author  gumau (3545) 
Entry type  Theorem 
Classification  msc 15A04 