= Spectral radius criterion for matrix powers
{title2=$G^k\to0\ \Longleftrightarrow\ \rho(G)<1$}
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 $\binom{k}{j}\lambda^{k-j}$, which tend to zero when $|\lambda|<1$; a zero-eigenvalue block becomes zero after finitely many powers. Conversely, an eigenvector gives $G^kv=\lambda^kv$, so convergence to zero excludes $|\lambda|\geq1$. This criterion underlies the convergence theorem for a <stationary iterative method for a linear system>.
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