Graduate Thesis Or Dissertation

 

Generalization of the Ostrowski-Schneider main inertia theorem Public Deposited

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  • As indicated by the title, this thesis generalizes the Main Inertia Theorem of Ostrowski and Schneider [8]. The first three results concern the formation of a polynomial function f(A, A*, H) so that the existence of an hermitian H for which f(A, A*, H) is positive definite is a necessary and sufficient condition that the matrix A have no eigenvalues on an arbitrary line, circle, and parabola (respectively) in the complex plane. The next two results are motivated by the existence of a hermitian H such that a certain polynomial function f(A, A*, H) being positive definite is necessary and sufficient that H have no eigenvalues on a certain point set in the complex plane. Finally, this thesis demonstrates how the methods previously described can generalize (1) the second part of the Main Inertia Theorem and (2) three theorems by Drazin and Haynsworth [5] concerning necessary and sufficient conditions for the existence of a set of m linearly independent eigenvectors of a complex matrix A all corresponding to real eigenvalues, purely imaginary eigenvalues, and eigenvalues of absolute value one respectively.
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