Mathematical Tools for Physics

7—Operators and Matrices 200

But this assumes that you already know the ratings of the sites, and that’s what you’re trying to find!
Write this in matrix language. Each site is an element in a huge column matrix{xi}.

xi=K

∑N

j=1

αijxj or





x 1

x 2

..

.



=K







0 0 1 0 1 ...

1 0 0 0 0 ...

0 1 0 1 1 ...

...











x 1

x 2

..

.





An entry of 1 indicates a link and a 0 is no link. This is an eigenvector problem, with the eigenvalue 1 /K. There
are many eigenvectors, but there is a constraint that lets you pick the right one. All thexis must be positive.
This algorithm is the key idea behind Google’s methods. They have gone well beyond this basic technique of
course, but the spirit of this method remains.