Upper bounds for the condition numbers of the GCD and the reciprocal GCD matrices in spectral norm
Let S={x1,,xn} S = { x 1 ,  , x n } be a set of n n distinct positive integers. The nn n  n matrix having the greatest common divisor (xi,xj) ( x i , x j ) of xi x i and xj x j as its i,j i , j -entry is called the greatest common divisor (GCD) matrix defined on S S , denoted by ((xi,xj)) ( ( x i , x j ) ) , or abbreviated as (S) ( S ) . The nn n  n matrix (S?1)=(gij) ( 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 S . In this paper, we present upper bounds for the spectral condition numbers of the reciprocal GCD matrix (S?1) ( S ? 1 ) and the GCD matrix (S) ( S ) defined on S={1,2,,n} S = { 1 , 2 ,  , n } , with n?2 n ? 2 , as a function of Eulers ? ? function and n n .
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