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

Boros and Rózsa (Linear Algebra Appl. 421 (2007) 407–416) reduced the singular value problem for the odd-order Sylvester–Kac matrix $P\in {\mathbb{R}}^{(2m+1)\times (2m+1)}$ to those for $Q^{(m)}\in {\mathbb{R}}^{(m+1)\times m}$ and $S^{(m)}\in {\mathbb{R}}^{m\times (m+1)}$ via a permutation similarity transformation. They noted that $(Q^{(m)}{)}^T{Q}^{(m)}-S^{(m)}(S^{(m)}{)}^T$ is positive definite, and also conjectured that ${\sigma }_i(S^{(m)})\ge {\sigma }_{i+1}(Q^{(m)})$ for $1\le i\le m-1$ and $m\ge 2$. In this paper, we prove that these inequalities are strict and establish the strict interlacing relation \begin{equation*} σ_1(Q^{(m)})>σ_1(S^{(m)})> \cdots>σ_m(Q^{(m)})> σ_m(S^{(m)})>0 \end{equation*} for all $m\ge 1$. Here, ${\sigma }_i(\cdot )$ denotes the $i$-th largest singular value of a matrix.