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

Abstract

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\(\frac{{n(n - 3)}}{2}\) in the symmetric caseSij=Sij. We give also lower bounds on the number of discrete ambiguities, the number of solutions being at least\(2^{\frac{{n(n - 3)}}{2}} \) in the first case and\(2^{\left[ {\frac{n}{2}} \right]\left[ {\frac{{n - 1}}{2}} \right] - 1} \) for the symmetric one, where [r] denotes the integral part ofr.