Linear Algebra By Kunquan Lan -fourth Edition- Pearson 2020 [ 95% PLUS ]

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

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.

We can create the matrix $A$ as follows:

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

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.

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

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_2 = A v_1 = \begin{bmatrix} 1/4 \

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

The converged PageRank scores are:

Using the Power Method, we can compute the PageRank scores as: The Google PageRank algorithm is a great example

The PageRank scores indicate that Page 2 is the most important page, followed by Pages 1 and 3.

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

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