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 $\{Y_nT_n[f]\}_n$, 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 $Y_n T_n[f]$ 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 $\{Y_nT_n[f]\}_n$ is distributed in the eigenvalue sense as $\pm |f|$. 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 $\phi_1$ 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.