Cyclic Decomposition Theorem

Every finite dimensional vector space can be written as a direct sum of cyclic invariant subspaces.

Theorem

• Let $T\in \mathcal{L}(V)$
• Let $f(x)=p_1(x{)}^{d_1}\dots p_k(T{)}^{d_k}$ be the characteristic polynomial
• Let $p(x)=p_1(x{)}^{r_1}\dots p_k(x{)}^{r_k}$ be the minimal polynomial Then, there exists ${\alpha }_1,\dots ,{\alpha }_j$ such that:

1. $V=Z({\alpha }_{1}T)\oplus \cdots \oplus Z({\alpha }_{j}T)$
2. $p=p_1$
3. $p_i$ divides $p_{i-1}$ for all $i\in \{2,\dots ,j\}$ Where $p_j$ is the minimal polynomial of $Z({\alpha }_{j}T)$ Note that by construction, $f=p_1{p}_2\dots p_j$

Theorems

• Cyclic Decomposition Theorem Lemma