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.
$V=\\bigoplus_{i=1}^{r}\\ker(p_{i}^{\\alpha_{i}}(T))$
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 $f=f_{1}\\ldots f_{r}$ and $(f_{i},f_{j})=1$ if $i\eq j$ , then

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

单词	PrimaryDecompositionTheorem
释义	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. $V=\\bigoplus_{i=1}^{r}\\ker(p_{i}^{\\alpha_{i}}(T))$ 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 $f=f_{1}\\ldots f_{r}$ and $(f_{i},f_{j})=1$ if $i\eq j$ , then 1. $V=\\bigoplus_{i=1}^{r}\\ker(f_{i}(T))$ 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.
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