The eigenvalue distribution of special $2$-by-$2$ block matrix sequences, with applications to the case of symmetrized Toeplitz structures

Abstract

Given a Lebesgue integrable function $f$ over $[0,2π]$, 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 flip permutation matrix, also called the anti-identity matrix. Because of the unitary character 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 the matrix $Y_n$. Under the assumption that the Fourier coefficients are real, we prove that $\{Y_nT_n[f]\}_n$ is distributed in the eigenvalue sense as \[ ϕ_g(θ)=\left\{ \begin{array}{cc} g(θ), & θ\in [0,2π], -g(-θ), & θ\in [-2π,0), \end{array} \right.\, \] with $g(θ)=|f(θ)|$. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as $ϕ_1$, under the mild assumption that $f$ is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.