Matrix inverse from a decaying matrix exponential ($A^{-1}=-{\int }_0^{\infty }{e}^{tA}dt$)

If every eigenvalue of a finite square matrix $A$ has negative real part, its matrix exponential decays and its integral converges. The Jordan normal form shows that polynomial factors in $t$ are dominated by exponential decay. Integrating $d(e^{tA})/dt=A{e}^{tA}$ gives $A{\int }_0^{\infty }{e}^{tA}dt=-I$, proving the displayed matrix inverse formula. In particular, for a Hermitian matrix with negative eigenvalues, the spectral theorem reduces the integral to scalar exponentials with integrals $-1/{\lambda }_j$. The minus sign is essential: the integral is positive definite in this Hermitian case, while $A^{-1}$ has negative eigenvalues.