spectral radius (Gelfand) formula

The spectral radius rho(A) is the largest absolute value among the eigenvalues of A: rho(A) = max |lambda_i|. It is the single number that decides whether repeatedly applying A makes vectors blow up or fade away. If rho(A) < 1 the powers A^k shrink to zero; if rho(A) > 1 some direction grows without bound.

Gelfand's formula expresses this purely through norms: rho(A) = lim as k -> infinity of ||A^k||^(1/k), and the limit is the same for every matrix norm. So even though a single ||A^k|| can mislead you, taking the k-th root and the limit strips away the norm choice and the transient behavior, leaving the true long-run growth rate.

The subtlety is that rho is not itself a norm — it can be zero for a nonzero matrix (any nilpotent matrix has all eigenvalues zero). What is always true is rho(A) <= ||A|| for every submultiplicative norm. The eigenvalues live inside every norm ball; Gelfand says they fill it exactly in the limit.

This matters everywhere powers appear: stability of iterative solvers like x_{k+1} = M x_k + c, the convergence of the Neumann series, and the long-term behavior of dynamical systems and Markov chains. The practical rule is simple — watch rho, not a one-shot norm.