Is there a way to algorithmically generate uniformly random determinant=1 matrices such that I have reasonably spanned the entire space of such matrices?
To elaborate more precisely: consider the same question for rotation matrices $R \in SO(2)$. I create $\theta_i = uniformrandom (-\pi/2 , \pi/2)$ and create $R_i = \big{[} \begin{matrix} cos \theta_i & -sin\theta_i\\sin\theta_i & cos\theta_i \end{matrix} \big{]} $ for i=1,2,...N
Now I know that for larger and larger N, I have spanned SO(2) more and more uniformly denser and that the likelyhood of missing out any "corners" reduces as I choose larger N. How do I mimic the exact same experiment for unit determinant matrices in general?, i.e. How to uniformly sample the space of unit determinant 2X2 matrices?, The "uniform" is essential :)