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Math Help - Compute the steady-state matrix of the stochastic matrix.

  1. #1
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    Question Compute the steady-state matrix of the stochastic matrix.

    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!)

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  2. #2
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    Re: Compute the steady-state matrix of the stochastic matrix.

    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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