I want to parametrize the set $\mathcal{U}_{1,-1}(2)$ of complex 2x2 unitary matrices with eigenvalues $\{1,-1\}$.
I know of various parametrizations for the set $\mathcal{U}(2)$ of complex 2x2 unitary matrices with arbitrary eigenvalues, and am looking for analogous parametrizations when the eigenvalues are $\{1,-1\}$.
My Ideas
I have already come up with two constructions for this, but they seem overly complicated (I hope they are correct at least):
- Take any parametrization $U_{\theta}$ of $\mathcal{U}(2)$. Then, $M_{\theta} := U_{\theta} \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} U_{\theta}^\dagger$ is a parametrization of $\mathcal{U}_{-1,1}(2)$.
- Take a parametrization $v_{\theta}$ of a complex 2D vector with norm $1$ (which one?). Compute $w_{\theta}$ orthogonal to $v_{\theta}$ (by swapping coordinates and switching the sign of the first dimension). Then, $M_{\theta,\varphi}:=\left[ \begin{array}{c|c} v_{\theta} & e^{i \pi \varphi} w_{\theta} \end{array}\right] \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \left[ \begin{array}{cc} v_\theta^\dagger \\ \hline e^{-i \pi \varphi} w_{\theta}^\dagger \end{array} \right]$ is a parametrization of $\mathcal{U}_{-1,1}(2)$, where $\varphi \in \mathbb{R}$.