Contrary to state feedback control, the design of static output feedback is quite challenging even for linear time- invariant state-space systems. In this paper we consider the eigenvalue placement from an algebraic point of view. Not all eigenvalues necessarily have to be placed, i.e., we consider a partial eigenvalue placement. In addition, the stability of the remaining dynamics must be ensured. We also exploit degrees of freedom in the controller design of multivariable systems in order to minimize suitable matrix norms of the feedback gain. The required computations are carried out using methods from algebraic geometry such as Gröbner bases and quantifier elimination.