# Thread: Primary Decomposition theory

1. ## Primary Decomposition theory

Dears,
Could you please help me in proving the following:

" 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

2. ## 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: