# The matrix exponential

## The matrix exponential function

Definition (The matrix exponential):

If $M$ is a square matrix then the exponential of $M$ is defined to be $\exp(M)=I+M+\frac{1}{2}M^{2}+\frac{1}{3!}M^{3}+\cdots=\sum_{n=0}^{\infty}\frac% {1}{n!}M^{n}.$ Here $M^{0}=I$ .

Example:

Let $M=\begin{pmatrix}0&-\theta\\ \theta&0\end{pmatrix}$ . We'll compute $\exp(M)$ . First let's compute the powers of $M$ :

• $M^{2}=\begin{pmatrix}-\theta^{2}&0\\ 0&-\theta^{2}\end{pmatrix}=-\theta^{2}I$ ,

• $M^{3}=\begin{pmatrix}0&\theta^{3}\\ -\theta^{3}&0\end{pmatrix}$ ,

• $M^{4}=(M^{2})^{2}=\theta^{4}I$ .

• $M^{5}=\theta^{4}M$ , etc

so we end up with: $\exp(M)=\begin{pmatrix}1-\frac{\theta^{2}}{2}+\frac{\theta^{4}}{4!}-\cdots&-% \theta+\frac{1}{3!}\theta^{2}-\cdots\\ \theta-\frac{1}{3!}\theta^{3}+\cdots&1-\frac{1}{2}\theta^{2}+\frac{1}{4!}% \theta^{4}-\cdots\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}$ where we have just observed that the power series in each entries are the Taylor series of cos and sin.

Remark:

This is the 2-by-2 matrix that gives you a rotation by an angle $\theta$ . You should imagine that, as in this example, the exponential map "eats" a very simple matrix (like an $n$ -by-$n$ antisymmetric matrix) and outputs a much more complicated and useful matrix (like an $n$ -by-$n$ rotation matrix).

## Pre-class exercise

Exercise:

Compute $\exp\begin{pmatrix}0&x\\ 0&0\end{pmatrix}$ and $\exp\begin{pmatrix}0&a&c\\ 0&0&b\\ 0&0&0\end{pmatrix}$ .