Inertia, positive definiteness and ${\mathrm{\ell }}_p$ norm of GCD and LCM matrices and their unitary analogs

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 ${\mathrm{\ell }}_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.