In this paper, we investigate the problem of the robust pole assignment of a family of matrices by the Lyapunov method. We use the properties of induced norms and matrix measures to examine whether or not all the eigenvalues lie in the desired region. We also give a necessary and sufficient condition for the nominally determined quadratic Lyapunov equation e(j theta)E*P+e(-j theta)PE<2I. With this necessary and sufficient condition, the poles of the system lying in the desired region can be inspected more efficiently. Finally, examples are given to show the feasibility of the method.