Is there a way to generate uniformly spaced, or better still, to parametrise Unitary Matrices close to Identity. The solution is required primarily for $SU(3)$, but also in general for $SU(n)$.
It might be relevant that I am trying to optimise a function that takes in a unitary matrix as its argument. I am aware of the fact that $SU(n)$ matrices are exponentiated hermitian traceless generators. But that does not quite help me.
The outline of the problem that I am trying to solve is as follows -
I have an equation for the evolution of some unitary matrix. The evolution is constrained by the unitarity of the matrix - and i have the fixed points of this flow by generating the entire spectrum of $SU(3)$ matrices. But now i also have to find out the stability of these fixed points. I would prefer to use a systematic root finding type algorithm to accomplish this - which is where generating unitary matrices close to the identity comes in.
After the initial brute force search in this space, i would also require increased resolution for bisection or Newton-Ralphson to work. I already have a way to generate a uniform distribution of unitary matrices RANDOMLY - refer here. But I cannot use that for the root finding bit.
Hope the explanation helps! I have heard something about using two Householder transformations to do this, but didnt find anything to back it up! Please cite references if you don't have enough time. I will complete the answers and choose the one that leads me to the correct answer.