Numerical methods for ordinary differential equations on matrix manifolds
Journal of Computational and Applied Mathematics
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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 characterization is in terms of the stationary points of a quadratic function on the unit sphere. This yields interlacing relationships between the eigenvalues of the original matrix and those of the perturbed matrix. As a result of the linkage between the perturbation and stationarity 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-cream cone, which in turn leads to a decomposition theorem for such matrices. In addition, results are obtained for nonsymmetric matrices regarding interlacing and majorization.