The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

Given a Lebesgue integrable function f over [−π, π], we consider the sequence of matrices {YnTn[f]}n, where Tn[f] is the n-by-n Toeplitz matrix generated by f and Yn is the anti-identity matrix. Because of the unitary nature of Yn, the singular values of Tn[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Yn. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn[f]}n is distributed in the eigenvalue sense as ±|f|. A generalization of this result to the block Toeplitz case is also shown. We also consider the preconditioning introduced by [J. Pestana and A. Wathen, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 273–288] and prove that the preconditioned matrix-sequence is distributed in the eigenvalue sense as φ₁ under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.