# Are the states (or set of states) absorbing, transient or recurrent?

• CTK
In summary, the matrix has no absorbing states and there are no transient states. The communication class is all of the state space.
CTK
Moved from technical forums so no template
Summary: Determine the absorbing states & communication classes of the given matrix.

Hello everyone,

If we have a state space of S = {1,2,3,4} and the following matrix:

\begin{bmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 1/3 & 2/3\\
1 & 0 & 0 & 0\\
0 & 1/2 & 1/2 & 0\\
\end{bmatrix}

Now, given the above, I don't think there are any absorbing states or sets of states, is that correct?
And since the above is a finite and irreducible closed set of states, then all of the states are recurrent and there are no transient states, right?
Also, since all the states communicate with each other, then the communication class is simply all of the state space, right?
Finally, not directly related to the above matrix, but just in general, if we don't have a limit law, then does that imply that we don't have a stationary distribution?

Please correct me if I am wrong, thanks.

Last edited:
That looks right. Did you draw a map of the states' interactions?

PeroK said:
That looks right. Did you draw a map of the states' interactions?
Yes, I did. But I am already struggling to type a matrix, let alone a graph ahahaha.
Just a quick follow up on the above matrix, what would the stationary distribution be for the above matrix in this case?
Also, what would the period of each state be? Because I got a period of 1 for every single state, but that's not possible because we just said above that it is periodic so it means the periodicity should be greater than 1, right?

Any suggestions would be appreciate it.

For a stationary distribution you are looking for a "back to front" eigenvector of the matrix.

Nothing looks periodic.

PeroK said:
For a stationary distribution you are looking for a "back to front" eigenvector of the matrix.

Nothing looks periodic.
I guess I have figured it out, thanks for your help, it is really appreciated. Have a good one.

CTK said:
I guess I have figured it out, thanks for your help, it is really appreciated. Have a good one.
There should be one stationary state - associated with the eigenvalue ##1## of course. By "back to front" eigenvector, I meant eigenvector of the transpose of the transition matrix. I've no idea why Markov theory has matrices acting on the right of row vectors - rather than the usual way round by acting from the left on column vectors.

PeroK said:
There should be one stationary state - associated with the eigenvalue ##1## of course. By "back to front" eigenvector, I meant eigenvector of the transpose of the transition matrix. I've no idea why Markov theory has matrices acting on the right of row vectors - rather than the usual way round by acting from the left on column vectors.

Yeah, in many cases, I am still not understanding the details but rather just implementing them to get an answer ahahaha. Thanks.

## 1. What is the difference between absorbing, transient, and recurrent states?

Absorbing states are states in a system where once entered, the system will stay in that state indefinitely. Transient states are states that a system will eventually leave and not return to. Recurrent states are states that a system will return to after leaving, possibly multiple times.

## 2. How can I determine if a state or set of states is absorbing?

A state or set of states is considered absorbing if the probability of transitioning to any other state is 0. This can be determined by looking at the transition matrix of the system.

## 3. Can a system have both absorbing and transient states?

Yes, a system can have both absorbing and transient states. This is common in systems with multiple states where some states are absorbing and others are not.

## 4. How can I identify recurrent states in a system?

Recurrent states can be identified by looking at the long-term behavior of a system. If a system returns to a state after leaving it, it is considered recurrent.

## 5. Why is it important to know if a state or set of states is absorbing, transient, or recurrent?

Knowing the absorbing, transient, and recurrent states in a system can help in understanding the long-term behavior of the system. It can also be useful in making predictions and decisions based on the probabilities of transitioning between states.

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