The $A$-optimal design problem and the $S$-matrix conjecture

Abstract

The $S$-matrix conjecture was formulated by Sloane and Harwit in 1976, motivated by an $A$-optimal design problem arising in spectroscopy. It states that if $A$ is a nonsingular real $n\times n$ matrix whose entries lie in $[0,1]$, then $${\Vert A^{-1}\Vert }_F\ge \frac{2n}{n+1}.$$ Moreover, equality holds if and only if $A$ is an $S$-matrix. In this paper, we prove the conjecture by combining a variational trace inequality in odd dimensions with a sharp range-constrained estimate for the Moore--Penrose inverse of a centered matrix in even dimensions.

arXiv comment25 pages. All comments are welcome!