Spectral properties of flipped Toeplitz matrices and related preconditioning

In this work, we investigate the spectra of "flipped" Toeplitz sequences, i.e., the asymptotic spectral behaviour of {Y_nT_n(f)}_n, where T_n(f)∈R^n×n is a real Toeplitz matrix generated by a function f∈L¹([-π,π]), and Y_n is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of Y_nT_n(f) are asymptotically described by a 2×2 matrix-valued function, whose eigenvalue functions are ±|f|. It turns out that roughly half of the eigenvalues of Y_nT_n(f) are well approximated by a uniform sampling of |f| over [-π,π]  while the remaining are well approximated by a uniform sampling of -|f| over the same interval. When f vanishes only on a set of measure zero, this motivates that the spectrum is virtually half positive and half negative. Some insights on the spectral distribution of related preconditioned sequences are provided as well. Finally, a wide number of numerical results illustrate our theoretical findings.