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

作者：Ipek Ahmet^*

来源：Computers & Mathematics with Applications, 2012, 63(3): 645-651.

DOI：10.1016/j.camwa.2011.11.016

• GCD matrices
• Matrix norms
• Euler's phi function

Let S = {x(1), . . . , x(n)} be a set of n distinct positive integers. The n x 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 x n matrix (S-1) = (g(ij)), where g(ij) = 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 phi function and n.

• 出版日期2012-2