spectral radius

The largest magnitude among a matrix's own eigenvalues -- the exact number that decides whether a linear iteration converges, no matter how it was built.

The spectral radius of a square matrix $T$ is $\rho(T)=\max_i\lvert\lambda_i\rvert$ , the largest magnitude among its eigenvalues $\lambda_i$ . For the linear fixed-point iteration $\mathbf{x}^{(k+1)}=T\mathbf{x}^{(k)}+\mathbf{c}$ , $\mathbf{x}^{(k)}\to\mathbf{x}^*$ from every starting guess if and only if $\rho(T)<1$ -- the necessary-and-sufficient condition diagonal dominance only ever guarantees one direction of.

Examples

A Jacobi iteration matrix with eigenvalues $0.5, -0.3, 0.1$ has spectral radius $0.5<1$ , so the iteration converges even if the original matrix isn't diagonally dominant.