Suppose that \(T : V \to V\) is a linear operator acting on a finite-dimensional vector space \(V\) over a field \(\mathbb{F}\). Additionally, suppose that the minimal polynomial of \(T\) is given by \(p(x) = \prod\limits_{q(x) \in S} \left ( q(x) \right )^{m_q}\), where \(S\) denotes the set of irreducible factors of \(p(x)\), and \(m_q\) denotes the multiplicity of \(q(x)\) as a factor of \(p(x)\) for all \(q(x) \in S\).

Then it is apparent that the set of polynomials \(\left \{ \left ( \frac{p}{q^{m_q}} \right)(x) : q(x) \in S \right \}\) is coprime.

\(\implies \sum\limits_{q(x) \in S} f_q(x) \left ( \frac{p}{q^{m_q}} \right )(x) = 1\) for some \(f_q(x) \in \mathbb{F}[x]\) for each \(q(x) \in S\)

[ by the analogue of Bézout’s identity for polynomials in \(\mathbb{F}[x]\) ].

Now, let \(E_q = f_q(T) \left ( \frac{p}{q^{m_q}} \right )(T)\) for each \(q(x) \in S\). Then:

  1. \[\sum\limits_{q(x) \in S} E_q = I\]

  2. For all \(q_1(x), q_2(x) \in S\) such that \(q_1(x) \neq q_2(x)\),

    \[E_{q_1} E_{q_2}\] \[= \left ( f_{q_1}(T) \left ( \frac{p}{q_{1}^{m_{q_1}}} \right )(T) \right ) \left ( f_{q_2}(T) \left ( \frac{p}{q_{2}^{m_{q_2}}} \right )(T) \right )\] \[= p(T) \left ( f_{q_1}(T) f_{q_2}(T) \left ( \frac{p}{q_{1}^{m_{q_1}} q_{2}^{m_{q_2}}} \right )(T) \right )\] \[= 0\]

    [ since \(p(x)\) is the minimal polynomial of \(T\) and hence \(p(T) = 0\) by definition ].

  3. For all \(q(x) \in S\):

    \[\operatorname{image}(E_q)\] \[= \{ u : u = E_q(v) \text{ for some } v \in V \}\] \[= \{ u : \left ( q(T) \right )^{m_q} (u) = (\left ( q(T) \right )^{m_q} E_q)(v) = p(T)(v) = 0 \text{ for some } v \in V \}\] \[= \operatorname{ker} \left ( q(T) \right )^{m_q}\] \[= \operatorname{ker} \left ( q(T) \right )^{\operatorname{dim}(V)}\]

    [ since the relative minimal polynomial of any vector in \(V\) must divide \(p(x)\), and \(m_q \leq \operatorname{dim}(V)\) ]

\[\implies V = \bigoplus\limits_{q(x) \in S} \operatorname{image}(E_q) = \bigoplus\limits_{q(x) \in S} \operatorname{ker} \left ( q(T) \right )^{\operatorname{dim}(V)}\]

The direct sum decomposition \(V = \bigoplus\limits_{q(x) \in S} \operatorname{ker}\left ( q(T) \right )^{\operatorname{dim}(V)}\) is said to be the primary decomposition of \(V\) relative to \(T\). It is useful to note that it also follows from 1., 2. and 3. that for all \(q(x) \in S\), \(E_q\) is a projection onto the component part \(\operatorname{ker} \left ( q(T) \right )^{\operatorname{dim}(V)}\) of the primary decomposition of \(V\) relative to \(T\).

Since \(V\) is arbitrary, it follows that finite-dimensional vector spaces have primary decompositions relative to linear operators acting on them in general, and moreover that projections onto component parts of primary decompositions of finite-dimensional vector spaces can also be found in general using the method discussed in this post.