Question: Let $P$ be a transition matrix. Suppose that
$$P^{k} =\begin{bmatrix}0.37 & 0.37 & 0.37\\0.33&0.33&0.33\\ 0.3&0.3&0.3\end{bmatrix}$$
As $k$ approaches infinity, if $x_k$ are state vectors and $x_0$ is a probability vector, then what does $x_k$ converges to?
My attempt: Since $Px=x$, we know that $(P – I_3)=x$. By subtracting $P$ with the identity matrix, I got:
$$(P – I_3) = x = \begin{bmatrix}-0.63&0.37&0.37\\ 0.33&-0.67&0.33\\ 0.3&0.3&-0.7\end{bmatrix}$$
For the next step, I generally find the null space of the matrix, which provide me with the info to find $x_1$, $x_2$, and $x_3$. But in this case, solving for the reduced row echelon form of the matrix is nearly impossible and I'm not really sure how I should approach this problem.
Can someone please show me how this is done? Thanks in advance
Best Answer
The steady state distribution is $(0.37,0.33,0.3)$ because for any distribution $x$, $Px$ is this distribution.