31 Matrix exponential $\exp(A)$ or expm (A)

Recalling the exponential series

\exp(z)=1+z+z^{2}/2!+\dots+z^{m}/m!+\dots,

for any $n\times n$ complex matrix $A$ , we define the matrix exponential by

\exp(A)=I+A+A^{2}/2!+\dots+A^{m}/m!+\dots.

Proposition (Wedderburn)

(i) For any $n\times n$ complex matrix $A$ , the exponential series converges.

(ii) $\exp(zA)\exp(wA)=\exp((z+w)A)$ for all $z,w\in{\bf C}$ ;

(iii) $\exp(zA)$ has inverse $\exp(-zA)$ for all $z\in{\bf C}$ ;

(iv) Let $\lambda$ be an eigenvalue of $A$ . Then $e^{z\lambda}$ is an eigenvalue of $\exp(zA)$ .

(v)

{{d}\over{dz}}\exp(zA)=A\exp(zA).