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

In this study we investigate the monotonic behavior of the smallest eigenvalue tntn and the largest eigenvalue TnTn of the n×nn×n matrix EnTEn, where the ij  -entry of EnEn is 1 if j|ij|i and 0 otherwise. We present a proof of the Mattila–Haukkanen conjecture which states that for every n∈Z+n∈Z+, tn+1≤tntn+1≤tn and Tn≤Tn+1Tn≤Tn+1. Also, we prove that limn→∞⁡tn=0limn→∞⁡tn=0 and limn→∞⁡Tn=∞limn→∞⁡Tn=∞ and we give a lower bound for tntn.