Markov transition matrix in canonical form?

In summary, a Markov transition matrix is a square matrix that represents the probabilities of transitioning from one state to another in a Markov chain. The canonical form of a Markov transition matrix is a diagonal matrix with the eigenvalues of the matrix on the main diagonal and the corresponding eigenvectors as columns. This form allows for easier analysis and calculation of long-term probabilities and steady-state distributions in a Markov chain. It is often used in various fields such as statistics, economics, and computer science to model and analyze systems with random and sequential behavior.
  • #1
cdotter
305
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As I understand, a Markov chain transition matrix rewritten in its canonical form is a large matrix that can be separated into quadrants: a zero matrix, an identity matrix, a transient to absorbing matrix, and a transient to transient matrix.

The zero matrix and identity matrix parts are easy enough, but I have no idea how to write transient to absorbing or transient to transient matrix. I've also found other sources that tell me to "rewrite the transition matrix so the transient states come first." I have no idea what this means.

I have also found an example, the drunken something or other problem, but the transition matrix is already written in its canonical form. This doesn't help me at all.

Could someone give me a dumbed down step-by-step guide, or maybe a worked example?
 
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  • #2
cdotter said:
As I understand, a Markov chain transition matrix rewritten in its canonical form is a large matrix that can be separated into quadrants: a zero matrix, an identity matrix, a transient to absorbing matrix, and a transient to transient matrix.

The zero matrix and identity matrix parts are easy enough, but I have no idea how to write transient to absorbing or transient to transient matrix. I've also found other sources that tell me to "rewrite the transition matrix so the transient states come first." I have no idea what this means.

I have also found an example, the drunken something or other problem, but the transition matrix is already written in its canonical form. This doesn't help me at all.

Could someone give me a dumbed down step-by-step guide, or maybe a worked example?

Your understanding seems faulty; there are many transition matrices that do not fit the pattern you describe. Why not read some standard sources, such as: http://www.cs.virginia.edu/~gfx/courses/2006/DataDriven/bib/texsyn/Chapter11.pdf ? This has numerous examples and has most of what you say you want.

RGV
 
  • #3
Nevermind, I figured it out. :)

http://www.aw-bc.com/greenwell/markov.pdf Example 7
 
  • #4
I didn't see your post before hitting reply. Thank you, I'll check that out as well.
 

Related to Markov transition matrix in canonical form?

1. What is a Markov transition matrix in canonical form?

A Markov transition matrix in canonical form is a square matrix that represents a discrete-time, finite-state Markov chain. Each element of the matrix represents the probability of transitioning from one state to another in one time step. The matrix is in canonical form when the sum of the probabilities in each row is equal to 1.

2. How is a Markov transition matrix in canonical form constructed?

A Markov transition matrix in canonical form is constructed by first identifying all the states in the Markov chain and assigning them as rows and columns in the matrix. Then, the transition probabilities are calculated and placed in the corresponding elements of the matrix. Finally, the probabilities in each row are normalized to ensure they sum to 1.

3. What is the significance of a Markov transition matrix in canonical form?

A Markov transition matrix in canonical form is significant because it allows for the analysis and prediction of the behavior of a Markov chain. It can be used to calculate the long-term probability of being in a particular state, as well as the expected number of time steps needed to transition between states.

4. Can a Markov transition matrix in canonical form have negative probabilities?

No, a Markov transition matrix in canonical form cannot have negative probabilities. The probabilities in each row must sum to 1, and negative probabilities would result in a sum greater or less than 1. In addition, negative probabilities do not make sense in the context of a Markov chain, as they represent the likelihood of transitioning from one state to another.

5. How is a Markov transition matrix in canonical form used in real-world applications?

A Markov transition matrix in canonical form is used in real-world applications to model and analyze various systems and processes. It is commonly used in the fields of economics, finance, biology, and engineering to study and predict the behavior of complex systems. It can also be used in machine learning and artificial intelligence algorithms for tasks such as speech recognition and natural language processing.

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