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

  • Teng Zhang
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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× n matrix whose entries lie in [0,1], then |A−1|F≥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.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026aoptimal,
  title = {{The $A$-optimal design problem and the $S$-matrix conjecture}},
  author = {Teng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20306},
  eprint = {2609.20306},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20306},
}
RIS
TY  - UNPB
TI  - The A-optimal design problem and the S-matrix conjecture
AU  - Teng Zhang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.20306
UR  - https://arxiv.org/abs/2609.20306
AB  - 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× n matrix whose entries lie in [0,1], then |A^−1|_F≥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.
ER  -