A Hurwitz matrix exponential has a polynomial-exponential decay bound

For complex $A$, its characteristic polynomial splits by [F7], so [F2] gives $A=UT{U}^{\ast }$ with $T$ upper triangular. For real $A$, regard its same real-entry matrix as acting on ${\mathbb{C}}^n$; its characteristic polynomial also splits by [F7], so [F2] gives a unitary Schur triangularization there. If $z=x+iy$ with real $x,y$, then $\mathrm{\parallel }Az{\mathrm{\parallel }}_2^2=\mathrm{\parallel }Ax{\mathrm{\parallel }}_2^2+\mathrm{\parallel }Ay{\mathrm{\parallel }}_2^2$, so the real and complex operator norms agree.

Write $T=D+N$, where $D$ contains its diagonal and $N$ is strictly upper triangular. By [F3] and [F4], the exponential series is differentiable in operator norm for each finite matrix. The identity $(UT{U}^{\ast }{)}^k=U{T}^k{U}^{\ast }$ in each series term gives $e^{tA}=U{e}^{tT}{U}^{\ast }$; [F5] gives $\mathrm{\parallel }U{\mathrm{\parallel }}_2=\mathrm{\parallel }{U}^{\ast }{\mathrm{\parallel }}_2=1$. Each diagonal exponential satisfies $\mathrm{\parallel }{e}^{sD}{\mathrm{\parallel }}_2\le e^{-\alpha s}$ for $s\ge 0$.

Differentiate $e^{(t-s)D}{e}^{sT}$ with respect to $s$ and integrate from $0$ to $t$. Since $T-D=N$, this gives $e^{tT}=e^{tD}+{\int }_0^t{e}^{(t-s)D}N{e}^{sT}ds$. Iterating this identity produces ordered-integral terms with $k$ factors $N$ and diagonal exponentials between them. The remainder after $n$ iterations contains $n$ strictly upper-triangular factors, with diagonal factors between them and an upper-triangular exponential at the end; every nonzero matrix-entry path would have to increase its index at least $n$ times in an $n\times n$ matrix, so this remainder is zero.

The ordered simplex for $k$ integration variables has volume $t^k\mathrm{/}k!$. By [F6], each $k$th term has norm at most $e^{-\alpha t}\mathrm{\parallel }N{\mathrm{\parallel }}_2^k{t}^k\mathrm{/}k!$. Summing the terms $0\le k<n$ proves the bound for $e^{tT}$; [F5] and step 2.1 transfer it to $e^{tA}$.