34 Proof of properties of exponential, concluded

(iii) Observe that, by (ii)

$$\exp (zA)\exp (-zA)=I=\exp (-zA)\exp (zA)$$

(iv) Let $v\in V$ satisfty $v\ne 0$ and $Av=\lambda v$. Then

$$\exp (zA)v=\sum _{j=0}^{\mathrm{\infty }}\frac{z^j{A}^{j}v}{j!}=\sum _{j=0}^{\mathrm{\infty }}\frac{z^j{\lambda }^{j}v}{j!}=e^{z\lambda }v.$$

(v) We consider (ii), and obtain as $h\to 0$

$$\frac{\exp ((z+h)A)-\exp (zA)}{h}=\exp (zA)\left(\frac{\exp (hA)-I}{h}\right)$$

$$=\exp (zA)\left(A+\frac{h{A}^2}{2!}+\frac{h^2{A}^3}{3!}+\mathrm{\dots }\right)\to \exp (zA)A$$