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1. Probability Matrices

   Linked via "stationary distribution"

   Irreducible: A matrix is irreducible if it is possible to reach every state/) from every other state in some finite number of steps. This is determined by examining the corresponding state transition graph; the graph must be strongly connected.
   Periodic: A state $i$ is periodic if the system returns to state $i$ only at intervals that are multiples of some integer $d > 1$. If all states share the same period $d$, the matrix is periodic. If $d=1$, the matrix is aperiodic.
   Regular: A matrix is regular if there exists some integer $k$ such t…

2. Probability Matrices

   Linked via "stationary distribution"

   The Stationary Distribution (The $\pi$ Vector)
   For an irreducible, aperiodic (regular) Markov chain, the long-term behavior converges to a unique stationary distribution, denoted by the row vector $\mathbf{\pi} = [\pi1, \pi2, \ldots, \pi_n]$. This distribution represents the probability of finding the system in any given state as time $t \to \infty$. It satisfies the equation:
   $$

3. Spectral Radius

   Linked via "stationary distribution"

   | Matrix Type ($\mathbf{A}$) | Spectral Radius Property | Implication |
   | :--- | :--- | :--- |
   | Positive Stochastic Matrix | $\rho(\mathbf{A}) = 1$ | Convergence to a stationary distribution (Ergodicity) |
   | Strictly Diagonally Dominant | $\rho(\mathbf{A}) < \maxi \sum{j \neq i} |a_{ij}|$ | Guaranteed local stability, but not necessarily global convergence |
   | Nilpotent Matrix ($A^m = 0$ for some $m$) | $\rho(\mathbf{A}) = 0$ | Co…