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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.

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    You can try generating an orthonormal base, which will then be the rows or columns of your unitary matrix.2012-04-17
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    What do you mean by *uniformly spaced*? Can't you use $\exp(-i\sum_j c_j H_j)$, where $H_j$ are the [Gell-Mann matrices](http://en.wikipedia.org/wiki/Gell-Mann_matrices) to parametrize your matrices?2012-04-17
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    @draks hope the edit answers your question!2012-04-17
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    @auke the problem is to generate an ensemble of unitary matrices for root finding, not just one!2012-04-17
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    @Debanjan This seems to be almost an exact duplicate of your last question (http://math.stackexchange.com/questions/129911/generating-unitary-matrices-numerically-close-to-the-identity-element/129945#129945), but with emphasis on $SU(3)$. I have posted a modification of my answer from last time below.2012-04-17
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    @JimBelk Last time I emphasized on generating the matrices **randomly**. This time however, I want to run a root-finding algorithm on this space. These are indeed separate problems - however you might want to post this impoved answer to the post [there](http://math.stackexchange.com/questions/129911/generating-unitary-matrices-numerically-close-to-the-identity-element).2012-04-17
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    What's the function you are trying to optimise?2012-04-17

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