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### Abstract

In this work, we investigate the spectra of "flipped" Toeplitz sequences, i.e., the asymptotic spectral behaviour of {Y_{n}T_{n}(f)}_{n}, where T_{n}(f)∈R^{n×n} is a real Toeplitz matrix generated by a function f∈L^{1}([-π,π]), and Y_{n} is the exchange matrix, with 1s on the main anti-diagonal. We show that the eigenvalues of Y_{n}T_{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_{n}T_{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.

Original language | English |
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Number of pages | 19 |

Journal | BIT Numerical Mathematics |

Publication status | Accepted/In press - 22 Nov 2018 |

### Keywords

- Toeplitz matrices
- spectral theory
- GLT theory
- Hankel matrices

## Fingerprint Dive into the research topics of 'Spectral properties of flipped Toeplitz matrices and related preconditioning'. Together they form a unique fingerprint.

## Projects

- 1 Finished

## Effective preconditioners for linear systems in fractional diffusion

EPSRC (Engineering and Physical Sciences Research Council)

19/01/18 → 19/06/20

Project: Research

## Cite this

*BIT Numerical Mathematics*.