On the parametrisation of unitary matrices by the moduli of their elements

The parametrisation of ann×n unitary matrix by the moduli of its elements is not a well posed problem, i.e. there are continuous and discrete ambiguities which naturally appear. We show that the continuous ambiguity is (n–1)(n–3)-dimensional in the general case and in the symmetric caseS _ij=S_ij. We give also lower bounds on the number of discrete ambiguities, the number of solutions being at least in the first case and for the symmetric one, where r] denotes the integral part ofr.