论文标题
会议矩阵中的光谱对称性
Spectral symmetry in conference matrices
论文作者
论文摘要
订单$ n $的会议矩阵是$ n \ times n $矩阵$ c $，带有对角线条目$ 0 $和偏高的条目$ \ pm 1 $满足$ cc^\ top =（n-1）i $。如果$ c $是对称的，则$ c $具有对称频谱$σ$（即$σ=-σ$）和eigenvalues $ \ pm \ pm \ sqrt {n-1} $。我们表明，$ c $的许多主要子膜也具有对称频谱，这导致了具有对称频谱的图形（或等效地，完全签名图的邻接矩阵）的Seidel矩阵示例。此外，我们表明某些具有对称光谱的Seidel矩阵可以通过这种结构来表征。
A conference matrix of order $n$ is an $n\times n$ matrix $C$ with diagonal entries $0$ and off-diagonal entries $\pm 1$ satisfying $CC^\top=(n-1)I$. If $C$ is symmetric, then $C$ has a symmetric spectrum $Σ$ (that is, $Σ=-Σ$) and eigenvalues $\pm\sqrt{n-1}$. We show that many principal submatrices of $C$ also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.
