Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but not sufficient that P has real eigenvalues.
Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but not sufficient that P has real eigenvalues.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 14EQ
Related questions
Question
![Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible
in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly
positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but
not sufficient that P has real eigenvalues.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f5ed8-56a6-4588-82ab-873ee102bd23%2F8ba7b8cf-5a04-49f2-8971-088abb1c7a30%2F8p1cqqs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Show that an irreducible Markov chain with a finite state space and transition matrix P is reversible
in equilibrium if and only if P = DS for some symmetric matrix S and diagonal matrix D with strictly
positive diagonal entries. Show further that for reversibility in equilibrium to hold, it is necessary but
not sufficient that P has real eigenvalues.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 4 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Elementary Linear Algebra (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305658004/9781305658004_smallCoverImage.gif)
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:
9781305658004
Author:
Ron Larson
Publisher:
Cengage Learning