Numerical diagonalization of a random hermitian matrix $H=U\Lambda U^{-1}$: enforce uniqueness and uniformity of $U$

Question

I've stumbled across this seemingly simple question, but I could not find a satisfactory answer. Suppose I have a complex hermitian random matrix $H$. It can be diagonalized by a unitary transformation, $H=U\Lambda U^{-1}$, however this decomposition is non-unique: $U_1\Lambda U_1^{-1}$ and $U_2\Lambda U_2^{-1}$ are equally legitimate if $U_1^{-1}U_2=\mathrm{diag}(e^{i\phi_1},\cdots,e^{i\phi_N})$, with $\phi_i$ arbitrary phases. To make the correspondence $H\to (\Lambda,U)$ one-to-one I have to restrict the unitary matrices to the coset space $U(N)/U(1)\otimes \cdots \otimes U(1)$. My question is: how can I enforce this restriction numerically? I mean, if I feed my matrix $H$ into any software performing numerical eigendecomposition, the algorithm will spit out one of the (infinitely many) diagonalizing matrices $U$ [chosen according to some arbitrary convention]. Therefore, the statistical properties of such $U$'s seem to be heavily dependent on the convention the software chooses, and devoided of any intrinsic meaning. But how can I 'tell' the software that the convention it chooses must be such that the corresponding matrix $U$ is uniformly (Haar) distributed in the unitary group (property that is seemingly implied by the restriction to the coset space, see e.g. http://arxiv.org/pdf/math-ph/0412017v2.pdf pag. 4-5)? Many thanks for your help.

Answer 1

A very partial answer.

1. You must only consider the generic hermitian matrices so that the eigenvalues are distinct.

2. According to your reference, it seems that the Haar measure can be decomposed in a product of one term relative to $\Lambda$ and one term relative to $U$. You speak about random matrices (when the $(H_{ij})_{i\leq j}$ are iid and follow the standard normal distribution ?) and you want that the corresponding matrix $U$ is uniformly (Haar) distributed in the unitary group; I am not sure that it is the case; the key constraint is the invariance by multiplication by unitary matrices. Anyway, it is eventually possible only if you fix $\Lambda$. Have you tried to choose a representative of $U$ that has a real diagonal? (it is easy to do).

3. About the distribution of the eigenvalues of $A$, you can see (in the real case) the well-known Wigner's result in http://mathworld.wolfram.com/WignersSemicircleLaw.html

About the phases of the eigenvalues of a random unitary matrix, see http://www.ams.org/journals/bull/2003-40-02/S0273-0979-03-00975-3/ where curious results are proved.