Working over $GF(2)$ for convenience.
Elementary matrices (identity with at most one additional off-diagonal coefficient) generate $SL_N$ (pretty generally, definitely over a field).
Suppose I sample elementary matrices choosing the potential off-diagonal term (which will be 1) by choosing the index $(i,j)$ uniformly and start multiplying $$ M_n=E_n\cdot E_{n-1}\cdot\ldots\cdot E_1. $$
Questions:
- Does $M_n$ converge to the uniform distribution on $SL_N$ as $n\to\infty$?
- If so, what is the rate of convergence, or more practically, how large must $n$ be for the product to look uniform?
What I really want to know is how to sample $SL_N$ ($=GL_N$ over $GF(2)$), using something like the above or otherwise to avoid something like rejection sampling of uniformly random matrices. That way I have a fixed number of iterations, can easily construct the inverse, etc.