The eigenvalue distribution of special $2$-by-$2$ block matrix sequences, with applications to the case of symmetrized Toeplitz structures
Provides a theoretical eigenvalue analysis for a class of structured matrices, relevant for numerical linear algebra and preconditioning, but the results are incremental extensions of known distribution results.
The paper derives the eigenvalue distribution of symmetrized Toeplitz matrices $Y_n T_n[f]$ and shows it is given by a specific function $\\phi_g$. It also proves that the preconditioned sequence by Pestana and Wathen is distributed as $\\phi_1$ under mild assumptions. Numerical experiments confirm the results.
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.