This Book, which consists of eight chapters, focuses on the interplay between some important problems, namely the controllability problem, the eigenvalue assignment problem and the partial eigenvalue assignment problem, arising in linear control system theory and their solution by recent results and methodologies in numerical linear algebra. Our objective is to study stabilizing control systems of different orders through the problem of reassigning a part of the spectrum of the systems by feedback control, leaving the rest of the spectrum invariant using different numerical methods. This problem is called the partial eigenvalue assignment problem. In chapter two, a simple method for generating a nonsingular solution of the matrix equation XH=HX. In chapter five, an explicit solution to the partial eigenvalue assignment problem of the linear control system. In chapter six, we propose an explicit solution to the partial eigenvalue assignment problem of damped gyroscopic second order control systems. In chapter seven, we propose an explicit solution to the partial eigenvalue assignment problem of high order control systems in single input case