Compute the steady-state matrix of the stochastic matrix.

**foz124** · April 29th 2012, 06:12 AM

3/5	0	0
0	1	1/8
2/5	0	7/8

Compute the steady-state matrix of the stochastic matrix. (Having trouble figuring this question out, could someone please help with how to solve this. Thanks!)

**Soroban** · April 29th 2012, 05:41 PM

Hello, foz124!

Your matrix is twisted . . . The rows must total 1.

$\begin{pmatrix}\frac{3}{5} & 0 & \frac{2}{5} \\ 0 & 1 & 0 \\ 0 & \frac{1}{8} & \frac{7}{8} \end{pmatrix}$

Compute the steady-state matrix of the stochastic matrix.

$\text{We want a column matrix }\,\begin{pmatrix}a\\b\\c\end{pmatrix}\,\text{ so that: }\:\begin{pmatrix}\frac{3}{5}&0&\frac{2}{5} \\ 0&1&0 \\ 0&\frac{1}{8}&\frac{7}{8}\end{pmatrix}\cdot\begin{ pmatrix}a\\b\\c\end{pmatrix} \;=\;\begin{pmatrix}a\\b\\c\end{pmatrix}$

$\text{This gives us three equations: }\:\begin{Bmatrix} \frac{3}{5}a + \frac{2}{5}c &=& a \\ b &=& b \\ \frac{1}{8}b + \frac{7}{8}c &=& c \end{Bmatrix}$

We find that the three equation are dependent.
So we need another equation . . . It is always: / $a + b + c \:=\:1$

Solving the system we get: . $\begin{Bmatrix}a &=& \frac{1}{3} \\ \\[-4mm] b &=& \frac{1}{3} \\ \\[-4mm] c &=& \frac{1}{3} \end{Bmatrix}$

Therefore, the steady-state matrix is: . $\begin{pmatrix}\frac{1}{3} \\ \\[-4mm] \frac{1}{3} \\ \\[-4mm] \frac{1}{3}\end{pmatrix}$

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