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

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.