Example

Here we give a few examples of circle mapping calculations. First let's take the real line $\mathbb{R}$. This corresponds to the equation $z=\bar{z}$, which translates to the Hermitian matrix $$H=\begin{bmatrix}0&i\\ -i&0\end{bmatrix}$$ The determinant is $-1$ Let's consider the Möbius transformation $T(z)$ corresponding to the matrix $\begin{bmatrix}1+i & -1\\ -1&1-i\end{bmatrix}$. According to the theory above, $T(C)$ is the circle or line corresponding to the matrix $$T(H)=\bar{\begin{bmatrix}1+i & -1\\ -1&1-i\end{bmatrix}^{-1}... ...d{bmatrix} \begin{bmatrix}1+i & -1\\ -1&1-i\end{bmatrix}^{-1}.$$ This simplifies to
$$\begin{align*} T(H)&=\bar{\begin{bmatrix}1-i&1\\ 1&1+i\end{bmatrix}}^T \begin{b... ...{bmatrix}\\ &= \begin{bmatrix}-2& -2-i\\ -2+i&-2\end{bmatrix}\\ \end{align*}$$
Since the upper left entry is nonzero, this is a circle. The center is $-b/a=-1-\frac12 i$, and the radius is $1/2$.