Spectral radius

The spectral radius of a square matrix $, denoted $\rho(A)$, is the maximum absolute value among its eigenvalues. It governs the asymptotic growth rate of matrix powers ^k$ and determines whether iterative methods converge. The spectral radius formula, $\rho(A) = \lim_{k \to \infty} \|A^k\|^{1/k}$, known as Gelfand's formula, reveals that the long-term behavior of repeated linear operations is controlled entirely by the largest eigenvalue magnitude. In networked dynamical systems, the spectral radius of the adjacency matrix determines whether linear contagion processes spread or die out.