spectral radius

The spectral radius of a matrix is the size of its largest eigenvalue measured by absolute value: rho(A) = max |lambda| over all eigenvalues lambda. It ignores direction and sign, asking only how big, in magnitude, the most dominant scaling factor is. It is a single number that captures the most aggressive stretching the operator can do along an eigen-direction.

Its central role is governing the long-term behavior of powers A^k. If rho(A) < 1, then A^k -> 0 as k grows (the dynamics decay); if rho(A) > 1, the powers blow up; if rho(A) = 1, they stay bounded or oscillate. More precisely, Gelfand's formula says rho(A) = lim ||A^k||^{1/k}, tying the spectral radius to the asymptotic growth rate of any matrix norm of the powers.

Although rho(A) <= ||A|| for every induced norm (the spectral radius never exceeds any operator norm), the spectral radius is generally NOT itself a norm — it can be 0 for a nonzero nilpotent matrix. It reappears throughout matrix analysis: convergence of iterative solvers, stability of discrete dynamical systems, and Perron-Frobenius theory for nonnegative matrices.