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arXiv · 1707.09473

Inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs

Pentti Haukkanen·László Tóth
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Abstract

Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.

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Pentti Haukkanen, László Tóth. 2017-07-29. Inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. https://arxiv.org/abs/1707.09473

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