= Nonnormal Fourier amplification matrix
{title2=$\sup_{\theta,nk\leq T}\|G(\theta)^n\|\leq C_T$}
For a system or a multilevel method, each <Fourier mode> evolves through an amplification matrix $G(\theta)$. Stability requires bounds on $G(\theta)^n$ uniform in frequencies and mesh parameters. Eigenvalue moduli alone do not suffice: $G=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ has unit eigenvalues but $G^n$ has off-diagonal entry $n$. Uniformly bounded eigenvector matrices, or a uniformly positive energy symmetrizer, can supply the missing power bound.
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