\[p_1 \left({x}\right), p_2 \left({x}\right), \ldots, p_r \left({x}\right)\]
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 \[A\]
.