95 Exponential and Inverses

Let $A$ be an $(n\times n)$ complex matrix. Then for $s>\parallel A\parallel$, the matrix $sI-A$ is invertible with inverse

$${(sI-A)}^{-1}={\int }_0^{\mathrm{\infty }}{e}^{-st}\exp (tA)𝑑t.$$

Proof. We have

$$(sI-A)\exp (t(A-sI))=-\frac{d}{dt}\exp (t(A-sI))$$

so

$${\int }_0^R(sI-A)\exp (t(A-sI))𝑑t={\int }_0^R-\frac{d}{dt}\exp (t(A-sI))dt$$

so by Fundamental Theorem of Calculus