# Thread: Calculating Stationary Distribution with Matrix

1. ## Calculating Stationary Distribution with Matrix

$
u_1 = 0.8u_1 + 0.6u_5

u_2 = 0.2u_1 + 0.4u_5

u_3 = 0.4u_2 + 0.4u_6

u_4 = 0.6u_2 + 0.6u_7

u_5 = 0.6u_3 + 0.6u_7

u_6 = 0.4u_3 + 0.4u_7

u_7 = 0.4u_4 + 0.2u_8

u_8 = 0.6u_4 + 0.8u_8

u_1 + u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8 + = 1
$

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?

2. Originally Posted by emterics90
$
u_1 = 0.8u_1 + 0.6u_5

u_2 = 0.2u_1 + 0.4u_5

u_3 = 0.4u_2 + 0.4u_6

u_4 = 0.6u_2 + 0.6u_7

u_5 = 0.6u_3 + 0.6u_7

u_6 = 0.4u_3 + 0.4u_7

u_7 = 0.4u_4 + 0.2u_8

u_8 = 0.6u_4 + 0.8u_8

u_1 + u_2 + u_3 + u_4 + u_5 + u_6 + u_7 + u_8 + = 1
$

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