Uniform random rotation matrix

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This section considers the generation of uniform random matrices `Q in RR^{n xx n}` such that `Q^T Q = I` and `det(Q) = 1`. In other words, we would like to generate a random matrix `Q in SO(n)` such that the distribution of `Q` is given by the Haar measure `P`:

for all measurable `A subset SO(n)`.

Rejection sampling

Let `Q` be a uniform random orthogonal matrix. Since `Q` is uniformly distributed in `O(n)`, it will have `det(Q) = -1` with probability 0.5. Thus, one way to generate rotation matrices with uniform distribution is to reject random orthogonal matrices until `det(Q) = 1`.

Reflection

Let `Q` be uniform random orthogonal matrix. If `det(Q) = -1`, one can get a `Q` with `det(Q) = 1` by negating the first column of `Q`. This seems to produce the same result as rejection sampling (when plotting the density of eigenvalues). However, I am lacking a proof for this. This is the current implementation.

Remark 1.7.5 - Page generated 16.06.2017 01:36.