Starting from the N-body friction matrix of an unconstrained system of N rigid particles immersed in a viscous liquid, we derive rigorous expressions for the corresponding friction and mobility matrices of a geometrically constrained dynamical system. Our method is based on the fact that geometrical constraints in a dynamical system can be cast in the form of linear constraints for the Cartesian translational and angular velocities of its constituents. Corresponding equations of motion for Molecular Dynamics simulations have been derived recently [1]. Using the concept of generalized inverse matrices, we find the form of the constrained friction and mobility matrix in Cartesian and in reduced coordinates. We show that the equations of motion for Stokesian Dynamics can be derived from a minimum principle which is similar to Gauß′ principle of least constraint in classical mechanics. We relate our approach for deriving constrained friction and mobility matrices to Kirkwood′s method where holonomic constraints acting between point-like particles are described by generalized coordinates and tensor algebra in curvilinear space. As an application, we perform a Stokesian Dynamics simulation of sedimentation of a small model polymer consisting of five spherical monomers connected by massless sticks and joints.