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The maximal asymptotic growth rate of a finite-dimensional periodic linear differential equation is $P^{-1}\log \rho (N)$, where $P$ is the coefficient period and $N$ the monodromy matrix of a periodic linear system. The spectral radius selects the dominant Floquet multiplier. Generic initial data attain this rate; data lying entirely in other invariant subspaces can grow or decay at different rates.

For a three-by-three matrix with diagonal entries one and every off-diagonal entry $\alpha$, the Jacobi method iteration matrix has eigenvalue $-2\alpha$ on the all-ones vector and eigenvalue $\alpha$ with multiplicity two on its orthogonal complement. Its spectral radius is $2|\alpha |$, so Jacobi iteration converges exactly when $|\alpha |<1/2$.

For every finite-dimensional normal matrix $A$, the matrix 2-norm equals the spectral radius:This follows by unitary diagonalization of a normal matrix and unitary invariance of the matrix 2-norm.