Defect of a unitary matrix

Abstract

We analyze properties of a map B = f(U) sending a unitary matrix U of size N into a doubly stochastic matrix defined by B_{i,j} = |U_{i,j}|^2. For any U we define its DEFECT, determined by the dimensionality of the space being the image Df(T_U Unitaries) of the space T_U Unitaries tangent to the manifold of unitary matrices Unitaries at U, under the tangent map Df corresponding to f. The defect, equal to zero for a generic unitary matrix, gives an upper bound for the dimensionality of a smooth orbit (a manifold) of inequivalent unitary matrices V mapped into the same image, f(V) = f(U) = B, stemming from U. We demonstrate several properties of the defect and prove an explicit formula for the defect of a Fourier matrix F_N of size N. In this way we obtain an upper bound for the dimensionality of a smooth orbit of inequivalent unitary complex Hadamard matrices stemming from F_N. It is equal to zero iff N is prime and coincides with the dimensionality of the known orbits if N is a power of a prime. Two constructions of these orbits are presented at the end of this work.