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arXiv · 1710.05243

Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic

Mauricio Barrera·Albrecht Boettcher·Sergei M. Grudsky·Egor A. Maximenko
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Abstract

It was shown in a series of recent publications that the eigenvalues of $n\times n$ Toeplitz matrices generated by so-called simple-loop symbols admit certain regular asymptotic expansions into negative powers of $n+1$. On the other hand, recently two of the authors considered the pentadiagonal Toeplitz matrices generated by the symbol $g(x)=(2\sin(x/2))^4$, which does not satisfy the simple-loop conditions, and derived asymptotic expansions of a more complicated form. We here use these results to show that the eigenvalues of the pentadiagonal Toeplitz matrices do not admit the expected regular asymptotic expansion. This also delivers a counter-example to a conjecture by Ekström, Garoni, and Serra-Capizzano and reveals that the simple-loop condition is essential for the existence of the regular asymptotic expansion.

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Mauricio Barrera, Albrecht Boettcher, Sergei M. Grudsky, Egor A. Maximenko. 2017-10-14. Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic. https://doi.org/10.1007/978-3-319-75996-8_2

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