Primary Decomposition and Invariant Subspace Question

Question

The question is from Hoffman & Kunze Section 6.8 on Primary Decomposition, number 10: Let T be a linear operator on the finite-dimensional space V, let $p = {p_1}^{r_1}\cdots{p_k}^{r_k}$ be the minimal polynomial for T, and let $V = W_1 \oplus\cdots\oplus W_k$ be the primary decomposition for T, i.e., $W_j$ is the null space of $p_j(T)^{r_j}$ . Let W be any subspace of V which is invariant under T. Prove that $$ W = (W \cap W_1) \oplus (W \cap W_2) \oplus \cdots\oplus (W \cap W_k) $$

Answer 1

Use the fact that the min poly of $T$ restricted to $W$ divides the min poly of $T$, to obtain min poly of $T|w$ as a product of powers of $p_i$'s, but here the powers are less or equal to $r_i$'s. Say the powers are $d_i$'s. Invoke Primary decomposition on W, and try proving ker $p_i(T)^d_i = W \cap W_i$.

Answer 2

The answer is in the text in the proof of the lemma on page 263.