I'm following Kobayashi's "Differential geometry of complex vector bundles". In section IV.2 (p. 107) we have the following data: An holomorphic vector bundle $(E,h)\longrightarrow (M,g)$ over a compact Kaehler manifold, a point $x\in M\,$ and the holonomy group $\Psi_x$ of the Chern connection. Then Koba says: Let \begin{equation} E_x= E_x^{(0)}+E_x^{(1)}+\cdots+E_x^{(k)} \end{equation} be the orthogonal decomposition of $E_x$ such that $\Psi_x$ is trivial on $E_x^{(0)}$ and irreducible on each of $E_x^{(1)},\cdots,E_x^{(k)}$ (Of course $E_x^{(0)}$ may be trivial).
My question is: How can I obtain this specific decomposition?
My first though was it is just some basic property of the holonomy group but I'm unfamiliar with this subject (holonomy group and representations). So, if someone can provide me with a reference tot his topic, I would appreciate it.