Parametrization of unitary matrices, parametrizing the symmetric $S$ matrix and the $NN$ $S$ matrix above the inelastic threshold

None of the $S$-matrix representations used in analyses of nucleon-nucleon ($NN$) scattering above the inelastic threshold is implicitly unitary, general, and connected to low-energy (Stapp) parameters, and hence correspondingly related to the physical properties of the scattering. Such a representation is also needed to solve the corresponding inverse scattering problem. I present an $S$-matrix representation which satisfies these requirements. As a by-product, I derive a parametrization of an $n$-dimensional unitary matrix.