Extreme singular values and eigenvalues of non-Hermitian block Toeplitz matrices
The asymptotic behaviour of singular values and eigenvalues of non-Hermitian block Toeplitz matrices \(\{T_n(f)\}\) generated by a multivariate matrix valued function \(f:Q^p\) \(\to\) \(\mathbb{C}\)\(^{h\times k}\), \(Q=(-\pi,\pi)\) is considered. It is proved that the interval of singular values is a proper cluster when \(f\) is continuous or belongs to a Klein algebra. Results for the minimal singular value of \(\{T_n(f)\}\) in the non-Hermitian case are discussed and second-order ergodic formulas and localization of eigenvalues of preconditioned matrices \(\{T^{-1}_n(g)T_n(f)\}\) are presented. The analysis of the number of outlying singular values is important to establish a priori bounds for the number of iterations we have to expect when a conjugate gradient method is used for normal equations.
scientific article; zbMATH DE number 1367893
scientific article