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I want to calculate a unique equation for the following linear equation. The background is the page rank algorithm where I want to find a vector solution for the Google matrix:

The matrix on which I've started is: $$\left(\begin{array}{rrrr} 1&13&1&1\\ 1&1&7&7\\ 13&1&1&1\\ 1&1&13&1\end{array}\right).$$

Which is multiplied by $1/16$ in the end, but I leave this out, since its easier to do calculations on the matrix without this scalar. So yes, I am looking for the left eigenvector with eigenvalue 1, however I want to find a matching solution with linear equation. I used the Gauss algorithm on the matrix and came to the following solution:

$a = a/16$

$b = b/16 - a/16$

$c = c/16 + a/16 - 14b/16$

$d = 17a/16/16 - 15b/8/16 + c/16/16 + d/16 $

(This equation is derived from a matrix by using the Gauss algorithm)

For me it is obivous, that there is no unique solution to this equation. I was, hoever, told, that there is one, when adding the following constraints:

$a + b + c + d = 1$ and $a,b,c,d \geq 0$

This what I actually want, but I don't see how this could help. Any hints or suggestions?

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    $b = b - a$ tells us $a = 0$, then we have $c = c - 14b$ which tells us $b = 0$, then we have $d = \frac{c}{16} + d$ which tells us $c = 0$ then we have $d = 1$, so we get $a = b = c = 0$, $d = 1$.2012-01-12
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    @DevenWare Sorry, I forgot that every value has to be multiplied with $1/16$ although I am not sure if this changes anything.2012-01-12
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    A matrix usually does have four eigenvectors (even if, like here, there is only one eigenvalue, $1$). The matrix corresponding to your system is defective, however, and there is thus only one set of $a,b,c,d$ values (only one eigenvector) that satisfy your equations.2012-01-12
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    Could you edit to explicitly show which values should be multiplied by $\frac1{16}$? Yes, it *might* change...2012-01-12
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    You changed the equations, which changes the answer, but I suspect these equations are still not what you want.2012-01-12
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    so I've added the original matrix2012-01-12

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