Abstract
A matrix \(A\) of size \(n\) is called \(g\)-circulant if \(A=[a_{(r-g s)\text { mod } n}]_{r,s=0}^{n-1}\), while a matrix \(A\) is called \(g\)-Toeplitz if its entries obey the rule \(A=[a_{r-g s}]_{r,s=0}^{n-1}\). In this note we study the eigenvalues of \(g\)-circulants and we provide a preliminary asymptotic analysis of the eigenvalue distribution of \(g\)-Toeplitz sequences, in the case where the numbers \(\{a_k\}\) are the Fourier coefficients of an integrable function \(f\) over the domain \((-\pi ,\pi )\): while the singular value distribution of \(g\)-Toeplitz sequences is nontrivial for \(g>1\), as proved recently, the eigenvalue distribution seems to be clustered at zero and this completely different behaviour is explained by the high nonnormal character of \(g\)-Toeplitz sequences when the size is large, \(g>1\), and \(f\) is not identically zero. On the other hand, for negative \(g\) the clustering at zero is proven for essentially bounded \(f\). Some numerical evidences are given and critically discussed, in connection with a conjecture concerning the zero eigenvalue distribution of \(g\)-Toeplitz sequences with \(g>1\) and Wiener symbol.
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We thank the anonymous referees for their valuable suggestions that helped us to improve the presentation of the results.
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Serra-Capizzano, S., Sesana, D. A note on the eigenvalues of \(g\)-circulants (and of \(g\)-Toeplitz, \(g\)-Hankel matrices). Calcolo 51, 639–659 (2014). https://doi.org/10.1007/s10092-013-0104-6
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DOI: https://doi.org/10.1007/s10092-013-0104-6