# Singular values of multiplicative Toeplitz matrices

Hilberdink, T. (2017) Singular values of multiplicative Toeplitz matrices. Linear and Multilinear Algebra, 65 (4). pp. 813-829. ISSN 1563-5139

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To link to this item DOI: 10.1080/03081087.2016.1204978

## Abstract/Summary

We study the asymptotic behaviour of the singular values of matrices with entries $a_{ij}=f(i/j)$ if $j|i$ and zero otherwise, with $f$ an arithmetical function. In particular, we study the case where $f$ is multiplicative and $F(x):=\sum_{n\leq x} |f(n)|^2$ is regularly varying. Our main result is that, under quite general conditions, the singular values are, asymptotically, $\sqrt{\mu_r F(n)}$, where $\{\mu_r:r=1,2,3,\ldots\}$ are the eigenvalues of some positive Hilbert-Schmidt operator.

Item Type: Article Yes Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics 66059 Taylor & Francis