On the Ilmonen-Haukkanen-Merikoski Conjecture

TL;DR

This paper proves the Ilmonen-Haukkanen-Merikoski conjecture, which identifies the minimal eigenvalue among a specific set of lower triangular (0,1)-matrices with ones on the diagonal, using spectral radius inequalities.

Contribution

The paper provides a rigorous proof of the IHM conjecture, establishing the minimal eigenvalue explicitly for a class of structured matrices.

Findings

Confirmed the IHM conjecture for all n.

Identified the matrix achieving the minimal eigenvalue.

Used spectral radius inequalities in the proof.

Abstract

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 \begin{equation*} c_n = \min_{Z\in L_n} \left\lbrace \mu_n^{(1)}(Z):\mu_n^{(1)} (Z) \text{ is the smallest eigenvalue of } Z \right\rbrace . \end{equation*} The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that $c_n$ is equal to the smallest eigenvalue of $Y_0Y_0^T$, where \begin{equation*} (Y_0)_{ij}=\left\lbrace \ \begin{array}{cl} 0 & \text{if } \ i<j, 1 & \text{if } \ i=j, \frac{1-(-1)^{i+j}}{2} & \text{if } \ i>j. \end{array} \right. \end{equation*} In this paper we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.