The Primary Decomposition Theorem

Let  
\[p_1 \left({x}\right), p_2 \left({x}\right), \ldots, p_r \left({x}\right)\]
  be distinct irreducible monic polynomials.
Let  
\[c \in K \setminus \left\{ {0}\right\}\]
  and  
\[a_1, a_2, \ldots, a_r, r \in \mathbb{Z}_{{}+{}}\]
  be constants.
Then
\[p \left({x}\right) = c p_1 \left({x}\right)^{a_1} p_2 \left({x}\right)^{a_2} \dotsm p_r \left({x}\right)^{a_r}\]

The primary decomposition theorem then states the following:
\[\quad \ker \left({p_i \left({T}\right)^{a_i} }\right)\]
  is a  
\[T\]
  invariant subspace of  
\[V\]
  for all  
\[i = 1, 2, \dotsc, r\]

\[ V = V_1 \oplus V_2 \oplus ... \oplus V_r\]

Notice that if  
\[A\]
  is a matrix associated with the linear transformation  
\[T\]
, and  
\[\mathbf{v}\]
  is an eigenvector associated with an eigenvalue  
\[\lambda_1\]
  then  
\[A \mathbf{v}= \lambda_1 \mathbf{v}_1\]
.
Suppose now that an eigenvalue  
\[\lambda_1\]
  gives rise to several eigenvectors. These eigenvectors are linearly independent and form a basis for the eigenspace  
\[V_1\]
, which is a subspace of  
\[V\]
, since if  
\[\mathbf{v}_1 , \mathbf{v}_2 , ..., \mathbf{v}_l \in V_1\]
  then  
\[A(a_1 \mathbf{v}_1 +a_2 \mathbf{v}_2 + ...+ a_l \mathbf{v}) =\lambda((a_1 \mathbf{v}_1 +a_2 \mathbf{v}_2 + ...+ a_l \mathbf{v}))\]

Hence each eigenspace is invariant under  
\[T\]
  and we given a linear operator  
\[T\]
  operating on a vector space  
\[V\]
, we can write  
\[V\]
  as a direct sum of the subs[paces generated by the eigenvectors of  
\[A\]
.
Since