= Negative spectra do not imply uniform semidiscrete stability
{title2=$L_h=\begin{pmatrix}-1&h^{-1}\\0&-1\end{pmatrix}$}
Every eigenvalue of this <nonnormal matrix> is $-1$, but $e^{tL_h}=e^{-t}\begin{pmatrix}1&t/h\\0&1\end{pmatrix}$. Applied to the second coordinate unit vector, its norm at $t=1$ is at least $1/(eh)$. Thus negative eigenvalues give no mesh-uniform semigroup bound by themselves. <Normal matrices>, uniformly conditioned eigenvector bases, or a direct dissipative energy estimate supply the missing control.
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