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• W2895568660 abstract "Given a Lebesgue integrable function $f$ over $[0,2pi]$, 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 [ phi_g(theta)=left{ begin{array}{cc} g(theta), & thetain [0,2pi], -g(-theta), & thetain [-2pi,0), end{array} right., ] with $g(theta)=|f(theta)|$. 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 $phi_1$, under the mild assumption that $f$ is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed." @default.
• W2895568660 created "2018-10-12" @default.
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• W2895568660 date "2018-10-08" @default.
• W2895568660 modified "2023-10-16" @default.
• W2895568660 title "The eigenvalue distribution of special $2$-by-$2$ block matrix sequences, with applications to the case of symmetrized Toeplitz structures" @default.
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• W2895568660 hasPublicationYear "2018" @default.
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