Chapter 6 Computing the Page Rank Vector

Question 1

When does the limit of H exist?

Answer 1

When J(H) has k blocks of size 1 x 1 if 1 occurs k times as an eigenvalue
When 1 is the only eigenvalue with absolute value 1 then all others are less than 1

Question 2

How can we tell the speed of convergence of H

Answer 2

Depends on How close to 1 the other eigenvalues are specifically the second largest. The further away from 1 the second largest eigenvalue is the faster it will converge

Question 3

Why do we care practically on the speed of convergence

Answer 3

Alot of calculation involved in calculating the powers of hat matrix. Each time we multiply H by itself it gets more expensive so we will want to know how quickly calculations can stop.

Question 4

What are the conditions imposed on S to have

limit = vw^transpose

Answer 4

S is m x m stochastic matrix with positive entries. So 1 is a simple eigenvalue fo S and V = (1,…,1)^transpose and w^transpose is a row eigenvector of S that is normalised

Question 5

How can we estimate the limit of a stochastic matrix and estimate the row eigenvector value

Answer 5

Square until we see the rows start to look alike to get an idea of the limit. the limit has structure of three rows of w^transpose.

Question 6

How does the jordan canonical form converge faster?

Answer 6

The smaller lamda is the quicker the convergence of Jr(lamda)^n to value of 0