$v_1 = A v_0 = \begin{bmatrix} 1/6 \ 1/2 \ 1/3 \end{bmatrix}$

Let's say we have a set of $n$ web pages, and we want to compute the PageRank scores. We can create a matrix $A$ of size $n \times n$, where the entry $a_{ij}$ represents the probability of transitioning from page $j$ to page $i$. If page $j$ has a hyperlink to page $i$, then $a_{ij} = \frac{1}{d_j}$, where $d_j$ is the number of hyperlinks on page $j$. If page $j$ does not have a hyperlink to page $i$, then $a_{ij} = 0$.

This story is related to the topics of Linear Algebra, specifically eigenvalues, eigenvectors, and matrix multiplication, which are covered in the book "Linear Algebra" by Kunquan Lan, Fourth Edition, Pearson 2020.

Using the Power Method, we can compute the PageRank scores as:

$A = \begin{bmatrix} 0 & 1/2 & 0 \ 1/2 & 0 & 1 \ 1/2 & 1/2 & 0 \end{bmatrix}$

The basic idea is to represent the web as a graph, where each web page is a node, and the edges represent hyperlinks between pages. The PageRank algorithm assigns a score to each page, representing its importance or relevance.

$v_0 = \begin{bmatrix} 1/3 \ 1/3 \ 1/3 \end{bmatrix}$

$v_2 = A v_1 = \begin{bmatrix} 1/4 \ 1/2 \ 1/4 \end{bmatrix}$

The converged PageRank scores are:

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.

Suppose we have a set of 3 web pages with the following hyperlink structure:

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.