On a conjecture of Ilmonen, Haukkanen and Merikoski concerning the smallest eigenvalues of certain GCD related matrices

Let Kn be the set of all n×n lower triangular (0,1)-matrices with each diagonal element equal to 1, Ln={YYT:Y∈Kn} and let cn be the minimum of the smallest eigenvalue of YYT as Y goes through Kn. The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that cn is equal to the smallest eigenvalue of Y0Y0T, where Y0∈Kn with (Y0)ij=1−(−1)i+j2 for i>j. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.