  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.
