MATH319 Slides 29 Functions of the matrix 31 Matrix exponential

\exp(A)

30 Cayley–Hamilton Theorem

Theorem

Let $A$ be a $n\times n$ complex matrix with characteristic polynomial $\chi_{A}(s)$ . Then

\chi_{A}(A)=0.

Proof See project topics.

Complex exponential For $z\in{\bf C}$ write $z=\Re z+i\Im z$ where $\Re z$ is the real part and $\Im z$ is the imaginary part. We define

\exp(z)=e^{z}=1+z+{{z^{2}}\over{2!}}+{{z^{3}}\over{3!}}+\dots,

which converges for all $z\in{\bf C}$ . We have $\exp(z+w)=\exp(z)\exp(w)$ and $(d/dz)\exp(z)=\exp(z)$ . Also, $e^{i\theta}=\cos\theta+i\sin\theta$ has $|e^{i\theta}|=1$ . Hence $e^{z}$ has modulus $|e^{z}|=e^{\Re z}$ and argument $\arg e^{z}=\Im z$ . In engineering, this is often called the phase.