## 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:
$\ker \left({p_i \left({T}\right)^{a_i} }\right)$
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
$A$
.