Eigenvalue locations of generalized companion predictor matrices

Generalized predictor companion matrices arise in the linear prediction approach for the fit of a weighted sum of n exponentials to a given set of data points. They are special solutions of matrix equations of the type H(l + p) S = H(l), where for each l greater than or equal to 0 H(l) is an M x N Hankel matrix obtained from this data (M greater than or equal to N > n). We discuss in this paper results about the eigenvalue locations of this class of solutions by means of linear algebra techniques. An application of these results in the case that all the exponents have either negative or positive real parts is that the n exponentials can correspond to eigenvalues which are outside the unit circle depending on the choice of generalized predictor companion matrices. The other (N - n) eigenvalues of these matrices always lie inside the unit circle and approach zero when p increases. This separation can facilitate their numerical calculation.