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

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

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

The converged PageRank scores are:

Page 1 links to Page 2 and Page 3 Page 2 links to Page 1 and Page 3 Page 3 links to Page 2

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

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 $a_{ij} = 0$. $v_0 = \begin{bmatrix} 1/3 \ 1/3 \ 1/3

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.

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. We can create a matrix $A$ of size