Canonical Decomposition Theorem

The idea of the canonical decomposition theorem is that, since intercommunication is an [[equivalence-relation]] we can use sets of intercommunicative states to partition the state space \(C\) into disjoint equivalence classes. #### Canonical Decomposition Theorem

Specifically, the state space \(S\) may be decomposed uniquely as \(S=T \cup C_{1} \cup C_{2} \cup \dots\) where:

1. \(T\) is the set of transient states; and
2. \(C_\alpha\) are closed disjoint irreducible classes of persistent states.

To put the transition matrix into canonical form, for each class \(C_{\alpha}\) obtain a transition matrix \(P_{\alpha}\) by only considering rows and columns from states in \(C_{\alpha}\), then after reordering the states the transition matrix \(P\) can be rewritten as \[ P=\begin{pmatrix}P_{1} & 0 & 0 & 0 & \cdots \\ 0 & P_{2} & 0 & 0 & \cdots \\ 0 & 0 & P_{3} & 0 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \\ Q_{1} & Q_{2} & Q_{3} & Q_{4} & \cdots\end{pmatrix} \] where \(Q_{\alpha}\) govern transitions for states in \(T\).