Rate of Convergence of Markov Processes

Distribution of states: $P_{1} = P_{0}M$, $P_{2} = P_{0}M^{2}$, $P_{l} = P_{0}M^{l}$

Eigenvectors $v$ of matrix $M$ satisfy $vM=\lambda v$, where $\lambda$ is the eigenvalue associated with eigenvalue $v$. I.e. $vM$ is parallel to $v$.

Repeated multiplication of $v$ by $M$ will cause it to shrink exponentially. $vM=\lambda v$, $vM^{2}=\lambda v M =\lambda^{2} v$, $vM^{l}=\lambda^{l} v$.

$P$ can be written as linear sum of its components ($x_{i}$) parallel to the eigenvectors. I.e. $P_{0} = \sum x_{i} v_{i}$

$P_{1}M = \sum x_{i} v_{i}M = \sum x_{i} \lambda_{i} v_{i}$
$P_{2}M = \sum x_{i} \lambda_{i} v_{i}M = \sum x_{i} \lambda_{i}^{2} v_{i}$
$P_{l}M = \sum x_{i} \lambda_{i}^{l} v_{i}$

Repeated multiplication of $P$ by $M$ will shrink components corresponding to small $\lambda$. I.e. except $\lambda=1$, are transients.

Eventually only left with component parallel to eigenvector with largest eigen value.