A note on the eigenvalues of -circulants (and of -Toeplitz, -Hankel matrices)

A matrix of size is called -circulant if , while a matrix is called -Toeplitz if its entries obey the rule . In this note we study the eigenvalues of -circulants and we provide a preliminary asymptotic analysis of the eigenvalue distribution of -Toeplitz sequences, in the case where the numbers are the Fourier coefficients of an integrable function over the domain : while the singular value distribution of -Toeplitz sequences is nontrivial for , 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 -Toeplitz sequences when the size is large, , and is not identically zero. On the other hand, for negative the clustering at zero is proven for essentially bounded . Some numerical evidences are given and critically discussed, in connection with a conjecture concerning the zero eigenvalue distribution of -Toeplitz sequences with and Wiener symbol.