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1. Spectral Radius

   Linked via "Jordan blocks"

   Convergence (Stability): If $\rho(\mathbf{A}) < 1$, the system is asymptotically stable, meaning $\mathbf{x}k$ approaches the zero vector irrespective of the initial state $\mathbf{x}0$ (assuming $\mathbf{x}_0$ is within the span of the generalized eigenvectors).
   Divergence (Instability): If $\rho(\mathbf{A}) > 1$, the system is unstable, and $\|\mathbf{x}_k\|$ generally grows without bound.
   Criticality: If $\rho(\mathbf{A}) = 1$, the system is marginally stable. Its behavior depends on the structure of th…