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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 greatestcommon divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ isdefined similarly. We consider inertia, positive definiteness and $\ell_p$ normof GCD and LCM matrices and their unitary analogs. Proofs are based on matrixfactorizations and convolutions of arithmetical functions.

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