# The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings

Mazza, M. and Pestana, J. (2021) The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings. SIAM Journal on Matrix Analysis and Applications, 42 (3). pp. 1319-1336. ISSN 0895-4798

In this work, we perform a spectral analysis of flipped multilevel Toeplitz sequences, i.e., we study the asymptotic spectral behavior of $\{Y_{{n}} T_{{n}} (f)\}_{{n}}$, where $T_{{n}}(f)$ is a real, square multilevel Toeplitz matrix generated by a function $f\in L^1([-\pi,\pi]^d)$ and $Y_n$ is the exchange matrix, which has 1's on the main antidiagonal. In line with what we have shown for unilevel flipped Toeplitz matrix sequences, the asymptotic spectrum is determined by a 2 x 2 matrix-valued function whose eigenvalues are $\pm |f|$. Furthermore, we characterize the eigenvalue distribution of certain preconditioned flipped multilevel Toeplitz sequences with an analysis that covers both multilevel Toeplitz and circulant preconditioners. Finally, all our findings are illustrated by several numerical experiments.