Stochastic Matrix

A stochastic matrix is a Square Matrix $P$ whose columns are Probability Vectors.

|det(P)| ⇐ 1, only volume contracting or preserving

Theorem as $k\to \infty$

$${\vec{x}}_{k+1}=P{\vec{x}}_k;k=0,1,2,...$$

If $P$ is a regular stochastic matrix, then $P$ has a unique steady-state vector $\vec{q}$, and $\overrightarrow{x_{k+1}}=P\overrightarrow{x_k}$ converges to $\vec{q}$ as $k\to \infty$; $(P^k\overrightarrow{x_0}{⟶}_{k\to \infty }\vec{q})$ where $P\vec{q}=\vec{q}$

Regular

• Stochastic matrix is regular if there $\exists (k\ge 1)P^k$ strictly has positive entries
• Regular $⟺$ unique steady state vectors
• Irregular $⟺$ $0\le n\ne 1$ steady state vectors