If you have a square probability transition matrix $P$ (as in a Markov process), then I understand there is a technique to finding $P^n$, although I am not entirely clear. I understand that if $P$ can be expressed as $NDN^{-1}$, where $D$ is a diagonal matrix consisting of the eigenvalues of $P$, then $P^n = ND^nN^{-1}$.
Are all of the eigenvalues $\lambda$ such that $0 \leq \lambda \leq 1$?
If so, the limit of $P^n$ as $n$ approaches infinity would not be very interesting, either each value would collapse to 0 or approach 1.
And if the eigenvalues are not in that range, it seems the results would not be probabilities.
Am I wrong or am I misunderstanding something about the technique?
No, unless $P$ happens to be symmetric it may have non-real eigenvalues, and even if symmetric it may have negative eigenvalues. However, all eigenvalues have absolute value $\le 1$ by the Perron-Frobenius theorem. If the Markov chain has period $k$, it will have eigenvalues that are $k$'th roots of unity. If it is aperiodic, the only eigenvalue of absolute value $1$ is $1$. In that case, the limit of $P^n$ as $n \to \infty$ exists.