On the S-matrix conjecture

TL;DR

This paper provides a concise proof of Cheng's 1987 result on the minimal Frobenius norm of the inverse of matrices with entries in [0,1], confirming the S-matrix conjecture for odd dimensions.

Contribution

It offers a simplified proof of Cheng's theorem, strengthening the understanding of matrix inverse norms in the context of the S-matrix conjecture.

Findings

Confirmed the minimal Frobenius norm for odd dimensions.

Provided a shorter proof of Cheng's 1987 result.

Strengthened the theoretical foundation of the S-matrix conjecture.

Abstract

Motivated with a problem in spectroscopy, Sloane and Harwit conjectured in 1976 what is the minimal Frobenius norm of the inverse of a matrix having all entries from the interval [0, 1]. In 1987, Cheng proved their conjecture in the case of odd dimensions, while for even dimensions he obtained a slightly weaker lower bound for the norm. His proof is based on the Kiefer-Wolfowitz equivalence theorem from the approximate theory of optimal design. In this note we give a short and simple proof of his result.