Murat Demirbüken’s research while affiliated with Bilkent University and other places

Let $K_n$ be the set of all $n\times n$ lower triangular (0,1)-matrices with each diagonal element equal to 1, $L_n = \{ YY^T: Y\in K_n\}$ and let $c_n$ be the minimum of the smallest eigenvalue of $YY^T$ as Y goes through $K_n$. The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that $c_n$ is equal to the smallest eigenvalue of $Y_0Y_0^T$, where $Y_0 \in K_n$ with $(Y_0)_{ij} = \frac{1-(-1)^{i+j}}{2}$ 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.