I could find many resources on generating random unitary matrices, usually citing [F. Mezzadri, *Notices of the AMS* 54 (2007), 592-604](http://arxiv.org/abs/math-ph/0609050) for a method which generates unitaries random with respect to the Haar measure, but none of them mentioned the special unitary subgroup. In particular, I'm not sure about one thing: say that we take random $n\times n$ unitaries $U\_n$ generated this way and construct special unitary matrices simply by $U\_n / \det(U\_n)^{1/n}$.


Are special unitary matrices generated this way also random with respect to the Haar measure?



Generating a random special unitary matrix
yes, this normalization will produce a uniform distribution in ${\rm SU}(N)$, however, it might be more efficient to generate directly random matrices with unit determinant (you'll need one fewer parameter and no need to calculate the determinant), as explained in [Composite parameterization and Haar measure for all unitary and special unitary groups](http://arxiv.org/abs/1103.3408)


