Matrix powers in finite precision arithmetic

If A is a square matrix with spectral radius less than 1 then $A^k \to 0\,{\text{as}}\,k \to \infty $, but the powers computed in finite precision arithmetic may or may not converge. We derive a sufficient condition for $fl( A^k ) \to 0\,{\text{as}}\,k \to \infty $ and a bound on $\| fl ( A^k ) \|$, both expressed in terms of the Jordan canonical form of A. Examples show that the results can be sharp. We show that the sufficient condition can be rephrased in terms of a pseudospectrum of A when A is diagonalizable, under certain assumptions. Our analysis leads to the rule of thumb that convergence or divergence of the computed powers of A can be expected according as the spectral radius computed by any backward stable algorithm is less than or greater than 1.