Localization of eigenvalues in elliptic regions

We prove that the eigenvalues λj of an n by n complex matrix A with its characteristic polynomial having real coefficients lie in the elliptic region defined by , where and This region is intersected with the strip to obtain an improved eigenvalue localization region. We also give bounds for the semiaxes, which can be computed without knowing the eigenvalues of A. When A has r < n nonzero eigenvalues, we obtain a smaller elliptic region containing such nonzero eigenvalues.