Inertia, positive definiteness and ℓp norm of GCD and LCM matrices and their unitary analogs

Let S={x1,x2,…,xn} 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×n matrix having f evaluated at the greatest common divisor of xi and xj as its ij entry. The LCM matrix [S]f is defined similarly. We consider inertia, positive definiteness and ℓp norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.