The structured distance to normality of banded Toeplitz matrices

Matrix nearness problems have been the focus of much research in linear algebra; see, e.g., [8,9,19,32]. In particular, characterizations of the algebraic variety of normal matrices and distance measures to this variety have received considerable attention; see [10,11,13,14,16,17,23,25–27]. Normal matrices are of interest because their eigenvalues are optimally conditioned and their singular values are the magnitude of the eigenvalues. Numerical methods for the computation of eigenvalues of Hermitian matrices are simpler than methods designed for the computation of eigenvalues of general matrices. In view of Corollary 2.1 below, the eigenvalues of a normal banded Toeplitz matrix of suitably restricted bandwidth can be determined by