= Euclidean semigroup growth bounds
{c}
{title2=$e^{ts(A)}\leq\|e^{tA}\|_2\leq e^{t\mu_2(A)}$}
The <spectral radius> of the <matrix exponential> gives the lower bound through the <spectral abscissa> $s(A)$. Differentiating the squared <Euclidean norm> along $q'=Aq$ and bounding its Hermitian quadratic form gives the upper bound through the <Euclidean logarithmic norm> $\mu_2(A)$. A <non-normal matrix> can have $\mu_2(A)>0$ despite $s(A)<0$, allowing <transient growth> before its ultimate decay.
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