Linear Algebra By Kunquan Lan -fourth Edition- Pearson 2020 Today

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.

The Google PageRank algorithm is a great example of how Linear Algebra is used in real-world applications. By representing the web as a graph and using Linear Algebra techniques, such as eigenvalues and eigenvectors, we can compute the importance of each web page and rank them accordingly.

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

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. Linear Algebra By Kunquan Lan -fourth Edition- Pearson 2020

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

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

Suppose we have a set of 3 web pages with the following hyperlink structure: Imagine you're searching for information on the internet,

We can create the matrix $A$ as follows:

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.

$A = \begin{bmatrix} 0 & 1/2 & 0 \ 1/2 & 0 & 1 \ 1/2 & 1/2 & 0 \end{bmatrix}$ By representing the web as a graph and

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.

$v_k = \begin{bmatrix} 1/4 \ 1/2 \ 1/4 \end{bmatrix}$

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.

The converged PageRank scores are: