Euler's identity in matrix form

Solution 1:

Perhaps more interesting, in the matrix context, is that if you look at $2 \times 2 $ matrices of the form $$ \begin{bmatrix} a & -b \\ b & a \end{bmatrix} $$ then they are in 1-1 correspondence with complex numbers, where $a + b\mathbf i$ corresponds to the matrix I've just written. Furthermore, addition of complex numbers and addition of matrices match under this correspondence, and so do multiplication of complex numbers and matrix multiplication. In other words, you could simply declare that this set of matrices is $\mathbf C$, since it has all the properties that $\mathbf C$ is supposed to have.

The nice thing is that the element $\mathbf i \in \mathbf C$ corresponds to the matrix $$ \mathbf {M_i} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$ (whose square is, indeed, the negative identity). You can then use matrix exponential to verify that $$\exp(\pi \mathbf {M_i}) = -\mathbf I,$$ as you'd expect, and this might be regarded as a kind of proof of Euler's identity.

Solution 2:

The rotation matrix in $\mathbb{R}^2$ can be written as \begin{equation} R(\theta)=\left(\begin{matrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\\ \end{matrix}\right)=\cos\theta \left(\begin{matrix} 1 & 0 \\ 0 & 1\\ \end{matrix}\right)+\sin\theta\left(\begin{matrix} 0 & -1\\ 1 & 0\\ \end{matrix}\right). \end{equation} If we denote \begin{equation} I=\left(\begin{matrix} 1 & 0\\ 0 & 1\\ \end{matrix}\right),~\mathbb{I}=\left(\begin{matrix} 0 & -1\\ 1 & 0\\ \end{matrix}\right), \end{equation} it is easy to check that $$I^2=I, I\mathbb{I}=\mathbb{I}I=\mathbb{I}, \mathbb{I}^2=-I,$$ and

\begin{eqnarray} e^{\mathbb{I}θ}&=&I+\mathbb{I}θ+\frac{1}{2!}(\mathbb{I}θ)^2+\frac{1}{3!}(\mathbb{I}θ)^3+\frac{1}{4!}(\mathbb{I}θ)^4+\frac{1}{5!}(\mathbb{I}θ)^5+\ldots\\ &=&\left(1-\frac{1}{2!}θ^2+\frac{1}{4!}θ^4-\ldots\right)I+\left(θ-\frac{1}{3!}θ^3+\frac{1}{5!}θ^5+\ldots\right)\mathbb{I}\\ &=&\cos\theta I+\sin\theta\mathbb{I} \end{eqnarray} or, $$e^{\mathbb{I}θ}=R(\theta).$$

Text Reference

Stillwell, J. Naive Lie Theory, Undergraduate Texts in Mathematics, Springer, 2008.