Consider the approximation of Navier-Stokes equations for a Newtonian fluid by Euler type systems with relaxation. This requires to decompose the second-order derivative terms of the velocity into first-order terms. We use Hurwitz-Radon matrices for this decomposition. We prove the convergence of the approximate systems to the Navier-Stokes equations locally in time for smooth initial data and globally in time for initial data near constant equilibrium states.