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Math Help - Calculating Stationary Distribution with Matrix

  1. #1
    Newbie emterics90's Avatar
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    Calculating Stationary Distribution with Matrix

    <br />
u_1 = 0.8u_1 + 0.6u_5<br /> <br />
u_2 = 0.2u_1 + 0.4u_5<br /> <br />
u_3 = 0.4u_2 + 0.4u_6<br /> <br />
u_4 = 0.6u_2 + 0.6u_7<br /> <br />
u_5 = 0.6u_3 + 0.6u_7<br /> <br />
u_6 = 0.4u_3 + 0.4u_7<br /> <br />
u_7 = 0.4u_4 + 0.2u_8<br /> <br />
u_8 = 0.6u_4 + 0.8u_8<br /> <br />
u_1 + u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8 + = 1<br />

    How can I solve the equations above using matrixes to get (u_1, u_2, ... , u_8)=\left(\frac{9}{34},\frac{3}{34},\frac{1}{17},  \frac{3}{34},\frac{3}{34},\frac{1}{17},\frac{3}{34  }, \frac{9}{34} \right)?

    I tried substitution but it soon became confusing trying to figure out which I had to substitute with which. Is there a systematic process I can use?
    Last edited by emterics90; February 7th 2006 at 01:42 AM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by emterics90
    <br />
u_1 = 0.8u_1 + 0.6u_5<br /> <br />
u_2 = 0.2u_1 + 0.4u_5<br /> <br />
u_3 = 0.4u_2 + 0.4u_6<br /> <br />
u_4 = 0.6u_2 + 0.6u_7<br /> <br />
u_5 = 0.6u_3 + 0.6u_7<br /> <br />
u_6 = 0.4u_3 + 0.4u_7<br /> <br />
u_7 = 0.4u_4 + 0.2u_8<br /> <br />
u_8 = 0.6u_4 + 0.8u_8<br /> <br />
u_1 + u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8 + = 1<br />

    How can I solve the equations above using matrixes to get (u_1, u_2, ... , u_8)=\left(\frac{9}{34},\frac{3}{34},\frac{1}{17},  \frac{3}{34},\frac{3}{34},\frac{1}{17},\frac{3}{34  }, \frac{9}{34} \right)?

    I tried substitution but it soon became confusing trying to figure out which I had to substitute with which. Is there a systematic process I can use?
    Gaussian elimination is a systematic method which will solve these.

    You can find an explanation on MathWorld or Wikipedia

    RonL
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