Spectral radius criterion for matrix powers ($G^k\to 0⟺\rho (G)<1$)

For a finite square matrix, powers tend to zero exactly when every eigenvalue has modulus less than one. In a complex Jordan normal form, the entries of a block power are sums of factors $\left(\genfrac{}{}{0ex}{}{k}{j}\right){\lambda }^{k-j}$, which tend to zero when $|\lambda |<1$; a zero-eigenvalue block becomes zero after finitely many powers. Conversely, an eigenvector gives $G^{k}v={\lambda }^{k}v$, so convergence to zero excludes $|\lambda |\ge 1$. This criterion underlies the convergence theorem for a stationary iterative method for a linear system.