Genelleşmiş yaklaşık spektrumun çoklu özdeğerlerle ilişkisi

Wilkinson studied the distance from a square matrix with distinct eigenvalues to the set of defective matrices in 1960s due to its connection with the sensitivity of eigenvalues. Malyshev derived a singular value optimization characterization for the distance. Recently, Alam and Bora established that the distance to defectiveness from a matrix corresponds to the smallest epsilon such that two components of the epsilon-pseudospectrum of the matrix coalesce. Our main aim is to generalize this relation between the distance to defectiveness and the pseudospectra. First we attempt to relate the algebraic characterization of Malyshev and geometric characterization of Alam and Bora. Then we focus on the main theme of this thesis, the distance to the set of matrices with a multiple eigenvalue of prescribed algebraic multiplicity, which we call generalized Wilkinson distance, and its geometric characterization in terms of pseudospectra. We introduce the generalized pseudospectrum as the set comprised of eigenvalues of prescribed multiplicity of all matrices within a given neighborhood. As a generalization of the work of Alam and Bora, we derive an upper bound for the generalized Wilkinson distance in terms of the coalescence of components of the generalized pseudospectra.