In this paper some new properties of the equality constrained and weighted least squares problem (WLSE) 122min||()||nxRWKxg∈−subject to Lx=h are obtained. We derive a perturbation bound based on an unconstrained least squares problem and deduce some equivalent formulae for the projectors for this unconstrained LS problem. We also present a new way to compute the minimum norm solution of the WLSE problem by using the QR, ULV or URV decompositions of the corresponding matrices and propose an algorithm to compute using the QR factorizations. Some numerical examples are provided to compare different methods for solving the WLSE problem.