# If $W$ is T-invariant, $W=(W\cap W_{1})\oplus\cdots\oplus (W\cap W_{k})$

Let $$T$$ an operator over a $$\mathbb{F}$$ vector space $$\mathbb{V}$$, with $$\dim(\mathbb{V})<\infty$$. Let $$p=p_{1}^{r_{1}}\cdots p_{k}^{r_{k}}$$ the minimal polynomial of $$T$$, and $$\mathbb{V}=W_{1}\oplus\cdots\oplus W_{k}$$ the primary decomposition of $$\mathbb{V}$$ for $$T$$.

Let $$\mathbb{W}$$ a subspace T-invariant of $$\mathbb{V}$$. I need to show that

$$\mathbb{W}=(\mathbb{W}\cap W_{1})\oplus\cdots\oplus (\mathbb{W}\cap W_{k})$$

I really don't know what to do. I will appreciate any tip.

• This is Exercise 10 of Section 6.8 in Linear Algebra, second edition, by Hoffman and Kunze. Dec 23, 2018 at 14:51

Let $$q$$ be the minimal polynomial of $$T | _W$$. Then $$q | p$$ because $$p(T|_W) = 0$$. Therefore $$q = {p_1}^{s_1} \cdots {p_k}^{s_k}$$ with $$0 \le s_k \le r_k$$. Now let $$W = U_1 \oplus \cdots \oplus U_k$$ be the primary decomposition of $$W$$. Clearly we have $$W = \left( U_1 \cap W \right) \oplus \cdots \oplus \left( U_k \cap W \right)$$. Now $$u_i \in U_i \cap W \iff {p_i(T|_W)}^{s_i} u_i = 0$$. (This follows directly from the proof of the Primary Decomposition Theorem.) But this implies (by a similar equivalence) that $$u_i \in W_i \cap W$$. Hence we have showed that $$W \subset \left( W_1 \cap W \right) \oplus \cdots \oplus \left( W_k \cap W \right)$$. The other inclusion is trivial hence we have proofed $$W = \left( W_1 \cap W \right) \oplus \cdots \oplus \left( W_k \cap W \right)$$.