regular Discrete Time Markov Chains

Question

I have a transition matrix $P$. I know that $P$ is regular if all $p^{(n)}_{ij}>0$ for some $n \geq 1$. Is there an algorithm that can help me to verify whether $P$ is regular without calculating $P^n, n=0,1, \ldots, \infty$? Thanks in advance.

Answer 1

Compute the eigenvalues of the transition matrix $P$. If the eigenvalue $1$ is unique then $P$ is regular, i.e. if the multiplicity of that eigenvalue is one then it is regular otherwise it is not regular.

You also have that if the transition matrix $P$ is irreducible and at least one entry of the main diagonal is nonzero, then it is regular. The markov chain is irreducible when all its states communicate with each other.

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edit:

My answer is not quite right but we decided to not delete it. Together with the comments it might help someone.

Read the comments.

"The crucial point is that $1$ is the only eigenvalue on the unit circle. The multiplicity of the eigenvalue $1$ being $1$ does not guarantee regularity (in any sense)" - Byron Schmuland