Distance problems for hermitian matrix polynomials an (epsilon) - pseudospectra based approach

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Definite and definitizable pencils and hyperbolic, quasihyperbolic and definite polynomi- als are Hermitian matrix polynomials with real eigenvalues of definite type that arise in many applications in science and engineering. Given a member of any of these classes, it is therefore of practical importance to know the distance to a nearest Hermitian polyno- mial outside the class. These distance problems are analysed with respect to a specific norm in the setting of Hermitian pseudospectra of these polynomials. Algorithms based on the bisection method are proposed for computing these distances and finding a nearest Hermitian polynomial outside the class. One of these algorithms computes the Crawford number which is the distance from a definite pencil to a nearest Hermitian pencil that is not definite. This algorithm also computes a nearest Hermitian pencil with a defective eigenvalue that attains the distance. The algorithms for computing the Crawford number and the solution of the distance problem for hyperbolic polynomials requires the computation of the smallest eigenvalue(s) of a positive definite matrix or the largest eigenvalue(s) of a negative definite matrix and corresponding eigenvectors at each step of the iteration. A homogeneous definition of eigenvalue type for eigenvalues of Hermitian polyno- mials on the extended real line is proposed. Properties of Hermitian pencils based on their canonical form under congruence are investigated. Properties of the Hermitian pseudospectra of regular Hermitian pencils and polynomials are also analysed with a view towards solving the distance problems. Several bounds on the Crawford number are proposed. Some of these relate the Crawford number to the distribution of the eigenvalues of the definite pencil. A pertur- bation bound on the eigenvalues of the definite pencil in terms of the Crawford number is also derived in the setting of the Hermitian pseudospectrum of the definite pencil..
Supervisor: S. Bora