# Primary Decomposition theory

• April 7th 2013, 07:51 PM
raed
Primary Decomposition theory
Dears,

" Let $T$ be a linear opearator on finite dimensional spsce $V$, let $p = p_1^{{r_1}} \cdots p_k^{{r_k}}$ be a minimal polynomial for $T$, and let $V = {W_1} \oplus \cdots \oplus {W_k}$ be the primary decomposition for $T$, i.e., $W_i$ is the null space of
${p_i}{(T)^{{r_i}}}$ . Let $W$ be any subspace of $V$ which is invarient under $T$. Prove that
$W = (W \cap {W_1}) \oplus (W \cap {W_2}) \ldots \oplus (W \cap {W_k})$ "

With Best Wishes
• April 10th 2013, 04:42 PM
johng
Re: Primary Decomposition theory
The canonical decomposition of a vector space with respect to a linear transformation T can be found in any good linear algebra book. The solution to your problem is really just understanding this decomposition:

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