Steady State Vector

A steady state vector for Stochastic Matrix $P$ is a Vector $\vec{q}$ such that $P\vec{q}=\vec{q}$.

• It is where the Markov Chain of $P$ converges after a sufficient amount of time.
• Solve for $(P-I)\vec{q}=0$ to find the steady state vector for $P$.
• $\vec{q}$ is defined as ${\lim }_{k\to \infty }\left(P^k\overrightarrow{x_0}\right)=\vec{q}$, also $P\vec{q}=\vec{q}$
• When you repeatedly apply $P$, the Subspace will approach the Span of…
  • An Eigenvector, if $P$ is Regular
  • A Set of Eigenvector with non-one Cardinality, if $P$ is irregular
    • If the cardinality is greater than 1, points in the subspace will converge to the nearest Eigenspace

Example

Determine the steady state vector for

$$P=\left[\begin{array}{cc}.8& .3\\ .2& .7\end{array}\right]$$

Goal: solve $P\vec{q}=\vec{q}$ $(P-I)\vec{q}=0$

$$\begin{array}{c}\left[\begin{array}{ccc}.8-1& .3& 0\\ .2& .7-1& 0\end{array}\right]\\ \left[\begin{array}{ccc}-.2& .3& 0\\ .2& -.3& 0\end{array}\right]\\ \left[\begin{array}{ccc}1& -\frac{3}{2}& 0\\ 0& 0& 0\end{array}\right]\end{array}$$ $$\therefore \vec{q}=t\left[\begin{array}{c}\frac{3}{2}\\ 1\end{array}\right]$$ $$\frac{3}{2}t+t=1$$ $$\therefore t=\left[\begin{array}{c}\frac{3}{5}\\ \frac{2}{5}\end{array}\right]$$