Eigenvalues of even very nice Toeplitz matrices can be unexpectedly erratic
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. Here we 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.
Keywords.Toeplitz matrix eigenvalue spectral asymptotics
Mathematics Subject Classification (2010).Primary 15B05 Secondary 15A18 41A60 47B35 65F15
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