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

2. $\ker (p_i^{{\alpha }_i}(T))$ is $T$-invariant for every $i$

3. If $T_i$ is the restriction of $T$ to $\ker (p_i^{{\alpha }_i}(T))$, 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. $V={\oplus }_{i=1}^r\ker (f_i(T))$

2. $\ker (f_i(T))$ 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.