Primary Decomposition Theorem

Theorem

• Let $T\in \mathcal{L}(V)$
• $dim(V)<\infty$
• Let $p(x)$ be the Minimal Polynomial with for $T$ with prime factorization $p(x)=p_1{x}^{r_1}\ast \cdots \ast p_k{x}^{r_k}$ (in other words, $p_j$ are distinct irreducible monic polynomials and $r_j\ge 1$)
• Let $W_j=ker(p_j{T}^{r_j})$ for $j\in 1,\dots ,k$ Then we get the properties:

1. $V=W_1\oplus \cdots \oplus W_k$
2. $W_i$ is $T$-invariant for all $i$
3. if $T_i$ is a restriction to $W_i$, then $P_i^{r_i}$ is the Minimal Polynomial of $T_i$

Properties

1. The minimal polynomial of each of the $T_i$ are a power of a single prime factor
2. If the characteristic polynomial has the same prime factors as the minimal polynomial, (i.e $f(x)=p_i(x{)}^{d_1}\ast \cdots \ast p_k(x{)}^{d_k}$) then we can characterize the dimension

Guides

• Primary Decomposition Theorem Example

Proof

Proving 1

• We show that $W_1+\cdots +W_k$ is a Direct Sum by showing that $W_i\cap W_j=\{0\}$ for all $i\ne j$
• Suppose $\alpha \in W_i$, and $\alpha \in W_j$. As $p_i$ and $p_j$ are Coprime, then we know there exists polynomial $a$, $b$ such that $a{p}_i+b{p}_j=1$
• This implies that $\alpha (T)p_i(T)+b(T)p_j(T)=I$
• So, $\alpha =I(\alpha )=\alpha (T)p_i(T{)}^{r_i}\alpha +b(T)p_j(T{)}^{r_j}\alpha$
• As $\alpha \in W_{i}i,W_j$, then: $\alpha =\alpha (T)0+b(T)0=0$
• …

Proving 2

• Note that $p_j(T)$ commutes with$T$
• It follows immediately that $W_j=ker(p_j(T{)}^{r_j})$ is $T$-invariant

Proving 3

• Let $T_j$ be the restriction of $T$ to $W_j$
• $p^{\prime }$ is the minimal polynomial of $T_j$
• Since $W_j=ker(p_j(T{)}^{r_j})$, $p_j(x{)}^{r_j}$ annihilates $W_j$
• This, $p^{\prime }|(p_j(x{)}^{r_j})$.
• This means that we can write $p_j=(p_j(x){)}^k$ for some $k\le r_i$