Publication Date:
2013-12-29
Description:
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.
Print ISSN:
0008-0624
Electronic ISSN:
1126-5434
Topics:
Mathematics
Permalink
