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1. Spectral Radius

   Linked via "Strictly Diagonally Dominant"

   | :--- | :--- | :--- |
   | 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 $m$) | $\rho(\mathbf{A}) = 0$ | Complete collapse of the system state after $m$ steps |