= Matrix inverse from a decaying matrix exponential
{title2=$A^{-1}=-\int_0^\infty e^{tA}\,dt$}
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 $d(e^{tA})/dt=Ae^{tA}$ 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>.
Back to article page
