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?