<s>[INST] Matrix exponential of a traceless $2\times2$ complex matrix How to calculate $\exp\left(t\begin{bmatrix}0 & z\\z^* & 0\end{bmatrix}\right)$, or $\exp\left(\begin{bmatrix}0 & v\\w & 0\end{bmatrix}\right)$ (where: $v, w \in \mathbb{C}$) in general?
 [/INST] Let $A=\pmatrix{0&v\\ w&0}$ and $x$ be any complex number such that $vw=-x^2$. By Cayley-Hamilton theorem, $A^2 = \operatorname{trace}(A)A-\det(A)I = vwI = -x^2I$. Therefore, when $vw\ne0$ (so that $x\ne0$),
\begin{align*}
e^A
&= I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\ldots\\
&= \left(I+\frac{A^2}{2!}+\frac{A^4}{4!}+\ldots\right)
+\left(A+\frac{A^3}{3!}+\frac{A^5}{5!}+\ldots\right)\\
&= \left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots\right)I
+\left(1-\frac{x^2}{3!}+\frac{x^4}{5!}-\ldots\right)A\\
&= \cos(x)I+\frac{\sin(x)}{x}A.\tag{1}
\end{align*}
Since the exponential function is continuous, when $vw=0$, we can take the limit of $(1)$ with $x\to0$ to obtain that $e^A= I+A$.
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