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A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices

  • Fuying Tang
  • Shuo Shi
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Abstract

Boros and Rózsa (Linear Algebra Appl. 421 (2007) 407--416) reduced the singular value problem for the odd-order Sylvester--Kac matrix P∈ℝ(2m+1)×(2m+1) to those for Q(m)∈ℝ(m+1)× m and S(m)∈ℝm×(m+1) via a permutation similarity transformation. They noted that (Q(m))TQ(m)−S(m)(S(m))T is positive definite, and also conjectured that σi(S(m))≥σi+1(Q(m)) for 1≤ i≤ m−1 and m≥2. In this paper, we prove that these inequalities are strict and establish the strict interlacing relation σ1(Q(m))>σ1(S(m))> ⋯>σm(Q(m))> σm(S(m))>0 for all m≥1. Here, σi(·) denotes the i-th largest singular value of a matrix.

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Cite

BibTeXRIS
BibTeX
@misc{tang2026proof,
  title = {{A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices}},
  author = {Fuying Tang and Shuo Shi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21702},
  eprint = {2610.21702},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21702},
}
RIS
TY  - UNPB
TI  - A proof of the interlacing conjecture for the singular values of Sylvester-Kac matrices
AU  - Fuying Tang
AU  - Shuo Shi
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21702
UR  - https://arxiv.org/abs/2610.21702
AB  - Boros and Rózsa (Linear Algebra Appl. 421 (2007) 407--416) reduced the singular value problem for the odd-order Sylvester--Kac matrix P∈ℝ^(2m+1)×(2m+1) to those for Q^(m)∈ℝ^(m+1)× m and S^(m)∈ℝ^m×(m+1) via a permutation similarity transformation. They noted that (Q^(m))^TQ^(m)−S^(m)(S^(m))^T is positive definite, and also conjectured that σ_i(S^(m))≥σ_i+1(Q^(m)) for 1≤ i≤ m−1 and m≥2. In this paper, we prove that these inequalities are strict and establish the strict interlacing relation σ_1(Q^(m))>σ_1(S^(m))> ⋯>σ_m(Q^(m))> σ_m(S^(m))>0 for all m≥1. Here, σ_i(·) denotes the i-th largest singular value of a matrix.
ER  -