One of the fundamental open problems in control theory is that of the stabilization of a linear time invariant dynamical system through static output feedback. We are given a linear dynamical system defined through \begin{align*} \mydot{w} &= Aw + Bu y &= Cw . \end{align*} The problem is to find, if it exists, a feedback $u=Ky$ such that the matrix $A+BKC$ has all its eigenvalues in the complex left half plane and, if such a feedback does not exist, to prove that it does not. Substantial progress has not been made on the computational aspect of the solution to this problem. In this paper we consider instead `which annihilating polynomials can a matrix of the form $A+BKC$ possess?'. We give a simple solution to this problem when the system has either a single input or a single output. For the multi input - multi output case, we use these ideas to characterize the annihilating polynomials when $K$ has rank one, and suggest possible computational solutions for general $K.$ We also present some numerical evidence for the plausibility of this approach for the general case as well as for the problem of shifting the eigenvalues of the system.