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 [-pi,pi], we consider the sequence of matrices {YnTn[f]}(n), where T-n[f] is the n-by-n Toeplitz matrix generated by f and Y-n, is the anti-identity matrix. Because of the unitary nature of Y-n, the singular values of T-n[f] and YnTn[f] coincide. However, the eigenvalues are affected substantially by the action of Y-n. Under the assumption that the Fourier coefficients of f are real, we prove that {YnTn, [f]}(n) is distributed in the eigenvalue sense as +/-vertical bar f vertical bar. 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 phi(1) 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.