Inertia, positive definiteness and ℓp\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 $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions

Quasi Kronecker products and a determinant formula

We introduce an extension of the Kronecker product for matrices which retains many of the properties of the usual Kronecker product. As an application we study matrices over divisor-closed sets with multiplicative entries, and show how these are quasi Kronecker products over the primes of simpler matrices. In particular this gives a formula for the determinant of such matrices which combines and generalizes a number of previous results on Smith type determinants

On the divisibility of meet and join matrices

AbstractLet (P,⩽)=(P,∧,∨) be a lattice, let S={x1,x2,…,xn} be a meet-closed subset of P and let f:P→Z+ be a function. We characterize the matrix divisibility of the join matrix [S]f=[f(xi∨xj)] by the meet matrix (S)f=[f(xi∧xj)] in the ring Zn×n in terms of the usual divisibility in Z, and we present two algorithms for constructing certain classes of meet-closed sets S such that (S)f divides [S]f. As an example we present the lattice-theoretic structure of all meet-closed sets with at most five elements possessing the matrix divisibility property. Finally, we show that our methods solve some open problems in the divisor lattice, concerning the divisibility of GCD and LCM matrices

Notes on the divisibility of GCD and LCM Matrices

Let S={x1,x2,…,xn} be a set of positive integers, and let f be an arithmetical function. The matrices (S)f=[f(gcd(xi,xj))] and [S]f=[f(lcm [xi,xj])] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f, respectively. In this paper, we assume that the elements of the matrices (S)f and [S]f are integers and study the divisibility of GCD and LCM matrices and their unitary analogues in the ring Mn(ℤ) of the n×n matrices over the integers