A proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix

作者：Altinisik Ercan^*; Buyukkose Serife

来源：Linear Algebra and Its Applications, 2015, 471: 141-149.

DOI：10.1016/j.laa.2014.12.020

• GCD matrix
• LCM matrix
• Eigenvalue
• Eigenvalue inequalities
• Matrix norm

In this study we investigate the monotonic behavior of the smallest eigenvalue t(n) and the largest eigenvalue T-n of the n x n matrix E-n(T) E-n, where the ij-entry of E-n is 1 if j vertical bar i and 0 otherwise. We present a proof of the Mattila-Haukkanen conjecture which states that for every n is an element of Z(+), t(n+1) <= t(n) and T-n <= Tn+1. Also, we prove that lim(n ->infinity) t(n) = 0 and lim(n ->infinity) T-n = infinity and we give a lower bound for t(n).

• 出版日期2015-4-15