Find the equilibrium vector for the transition matrix below

Find the equilibrium vector for the transition matrix below

215 points

Find the equilibrium vector for the transition matrix below

$$\begin{bmatrix}0.4 & 0.4 & 0.2\\0.2 & 0.2 & 0.6 \\ 0.4 & 0.3 & 0.3\end{bmatrix}$$

Find the
homework

S
9.9k points

Suppose $V=[v_1\,v_2\,\,v_3]$ is the equilibrium vector. To find V solve the matrix equation $VP=V$ and use the fact that the sum of the entries of V must be 1,

$$VP=V \longrightarrow$$

$$[v_1\,v_2\,\,v_3] \begin{bmatrix}0.4 & 0.4 & 0.2\\0.2 & 0.2 & 0.6 \\ 0.4 & 0.3 & 0.3\end{bmatrix} = [v_1\,v_2\,\,v_3]$$

this gives us the system of equations,

$$\left\{ \begin{array}{c} v_1+v_2+v_3=1 \\ 0.4v_1+0.6v_2+0.4v_3 = v_1 \\ 0.4v_1+0.2v_2+0.3v_3 = v_2 \\ 0.2v_1+0.2v_2+0.3v_3 = v_3 \end{array} \right. \longrightarrow$$

$$\left\{ \begin{array}{c} v_1+v_2+v_3=1 \\ -0.6v_1+0.6v_2+0.4v_3 = 0 \\ 0.4v_1-0.8v_2+0.3v_3 = 0 \\ 0.2v_1+0.2v_2-0.7v_3 = 0 \end{array} \right.$$

solving this system gives,

$$v_1=\frac{25}{54}$$ $$v_2=\frac{17}{54}$$ $$v_3=\frac{2}{9}$$

with transition matrix $\big[\frac{25}{54} \, \frac{17}{54} \,\,\frac{2}{9}\big]$

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