MORE RESULTS ON EIGENVECTOR SADDLEPOINTS AND EIGENPOLYNOMIALS.

This paper extends the author's earlier work which proved that, for a certain class of Hermitian eigenproblems AP = λBP (including, most notably, the Toeplitz eigenproblem), the eigenvectors can be represented as saddlepoints of a special form of the Rayleigh quotient. Each saddlepoint solves a min-max/max-min optimization problem whose optimal value is the eigenvalue. The zeros of the eigenpolynomial factors are located with respect to the unit circle. These results were proved, in part, by using an inertia theorem for Stein equations. For another class of Hermitian eigenproblems AP = λBP, an analogous set of results are derived here using an inertia theorem for Lyapunov equations. In this case, the zeros of the eigenpolynomial factors are located with respect to the imaginary axis, and sufficient conditions for the eigenpolynomial factors to have only extended imaginary zeros are established. The highlight of the article is an important theorem that yields a general factorization of saddlepoint eigenpolynomials and parameterizes all eigenvector saddlepoint representations for an m-dimensional eigenspace. Several key extensions to the theory are also presented. [ABSTRACT FROM AUTHOR]