TRUST REGION PROBLEMS AND NONSYMMETRIC EIGENVALUE PERTURBATIONS

A characterization is given for the spectrum of a symmetric matrix to remain real after a nonsymmetric sign-restricted border perturbation, including the case where the perturbation is skew-symmetric. The chara cterization is in terms of the stationary points of a quadratic functi on on the unit sphere. This yields interlacing relationships between t he eigenvalues of the original matrix and those of the perturbed matri x. As a result of the linkage between the perturbation and stationarit y problems, new theoretical insights are gained for each. Applications of the main results include a characterization of those matrices that are exponentially nonnegative with respect to the n-dimensional ice-c ream cone, which in turn leads to a decomposition theorem for such mat rices. In addition, results are obtained for nonsymmetric matrices reg arding interlacing and majorization.