Imagine you're searching for information on the internet, and you want to find the most relevant web pages related to a specific topic. Google's PageRank algorithm uses Linear Algebra to solve this problem.
$v_2 = A v_1 = \begin{bmatrix} 1/4 \ 1/2 \ 1/4 \end{bmatrix}$ Linear Algebra By Kunquan Lan -fourth Edition- Pearson 2020
We can create the matrix $A$ as follows: Imagine you're searching for information on the internet,
The PageRank scores are computed by finding the eigenvector of the matrix $A$ corresponding to the largest eigenvalue, which is equal to 1. This eigenvector represents the stationary distribution of the Markov chain, where each entry represents the probability of being on a particular page. we can use the Power Method
Suppose we have a set of 3 web pages with the following hyperlink structure:
$v_k = \begin{bmatrix} 1/4 \ 1/2 \ 1/4 \end{bmatrix}$
To compute the eigenvector, we can use the Power Method, which is an iterative algorithm that starts with an initial guess and repeatedly multiplies it by the matrix $A$ until convergence.