Chapter 5 Perron Frobenius Theorem

Question 1

What is the adjacency matrix

Answer 1

0, 1 matrix

Question 2

What is the H matrix

Answer 2

Obtained by normalising the rows of the adjacency matrix and so is non negative with sums of 1 on rows

Question 3

State the perron frobenius theorem

Answer 3

A is n x n with all entries non negative then:

1. A had real eigenvlaue p, p>=|lamda| for every eigenvalue of A
2. There is eigenvector v with non negative entries and row eigenvector W ^transpose corresponding the the eigenvalue p
3. If A had positive entries P is simple. furthermore p>|lamda| for all eigenvalues lamda not qual to p of A

Question 4

Why do we have the perron frobenius theorem

Answer 4

For really big matrices we cannot necessarily find the jordan blocks or eigenvalues to tell me about the limit

Question 5

What does a simple eigenvalue mean

Answer 5

It is not repeated

Question 6

If A is positive then how is p simple?

Answer 6

As eigenvalue of A and vectors v and w have positive entries

Question 7

What is the name given to eigenvalue p in this theorem

Answer 7

perron eigenvalues

Question 8

What is the name given to the nonnegative eigenvector v

Answer 8

Perron eigenvector

Question 9

What is a key thing to remember when finding the left and right perron eigenvectors

Answer 9

They are positive typically so we should choose their values accordingly. Can be non negative if A is also non negative.

Question 10

What is a stochastic matrix

Answer 10

An n x n matrix is stochastic if all entries are non negative and each row sum is 1

Question 11

How does the perron frobenius theorem relate to stochastic matrix

Answer 11

Then 1 is the perron eigenvalue of S and e = (1,1,1….1) is the corresponding perron eigenvector and we have |lamda|<=1 for all other eigenvalues

Question 12

What does column stochastic mean

Answer 12

Matrix A is column stochastic if all entries are non negative and each column sum of A is 1. 1 is a left perron eigenvector of a column stochastic A

Question 13

What is doubly stochastic matrix

Answer 13

Non negative matrix that is both stochastic and column stochastic

Question 14

What does perron frobenius tell us about a stochastic matrix with all positive entries

Answer 14

Perron eigenvalue is 1 and all other eigenvalues are less than 1. It has right eigenvector e= (1,1,1,…,1)

Question 15

When does the limit exist for a stochastic matrix

Answer 15

Only when it is positive as then only 1 eigenvalue is equal to 1

Question 16

What is the purpose of the google matrix in finding the page rank

Answer 16

Matrix H is used in the page rank calculation but it doesnt necessarily have all positive entries so limit may not exist. It is replaced with modified G matrix which is stochastic with all positive entries.