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 $k$, $\dim V=n$, and $T\colon V\to V$ a linear operator, such that its minimal polynomial (or its annihilator polynomial) is $m_{T}$, which decomposes in $k[X]$ into irreducible factors as $m_{T}=p_{1}^{\alpha_{1}}\ldots p_{r}^{\alpha_{r}}$. Then,

1. 1.

$V=\bigoplus_{i=1}^{r}\ker(p_{i}^{\alpha_{i}}(T))$

2. 2.

$\ker(p_{i}^{\alpha_{i}}(T))$ is $T$-invariant for every $i$

3. 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 $f=f_{1}\ldots f_{r}$ and $(f_{i},f_{j})=1$ if $i\neq j$, then

1. 1.

$V=\bigoplus_{i=1}^{r}\ker(f_{i}(T))$

2. 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.

Title primary decomposition theorem PrimaryDecompositionTheorem 2013-03-22 14:15:30 2013-03-22 14:15:30 gumau (3545) gumau (3545) 8 gumau (3545) Theorem msc 15A04