13 - Primary Decomposition Theorem

Primary Decomposition Theorem

Let $m_T$ be the minimal polynomial and write it in the form such that

$$m_T(x)=f_1^{q_1}(x)\cdots f_r^{q_r}$$

where $f_i$ are the distinct monic irreducible polynomials

Define $W_i:=\mathrm{Ker}(f_i^{q_i}(T))$ then

$$\begin{array}{rl}(i)& V=W_1\oplus \cdots \oplus W_r\\ (ii)& W_i\text{ is }T-\text{invariant}\\ (iii)& m_{T{\mid }_{W_i}}=f_i^{q_i}\end{array}$$

Proof

Put $a=f_1^{q_1}\cdots f_{r-1}^{q_{r-1}}$ and $b=f_r^{q_r}$
Proceed on induction on $r$ using Decomposition Theorem of a Vector Space