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

Amatrix A of size n is called g-circulant if A = [a(r−gs) mod n]n−1r,s=0, while a matrix A is called g-Toeplitz if its entries obey the rule A = [ar−gs]n−1r,s=0. In this notewe 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 {ak } are the Fourier coefficients of an integrable function f over the domain (−π, π): 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.