Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm

Abstract

Let S = { x 1 , … , x n } be a set of n distinct positive integers. The n × n matrix having the greatest common divisor ( x i , x j ) of x i and x j as its i , j -entry is called the greatest common divisor (GCD) matrix defined on S , denoted by ( ( x i , x j ) ) , or abbreviated as ( S ) . The n × n matrix ( S − 1 ) = ( g i j ) , where g i j = 1 ( x i , x j ) , is called the reciprocal greatest common divisor (GCD) matrix on S . In this paper, we present upper bounds for the spectral condition numbers of the reciprocal GCD matrix ( S − 1 ) and the GCD matrix ( S ) defined on S = { 1 , 2 , … , n } , with n ≥ 2 , as a function of Euler’s ϕ function and n .