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 \(\Sigma \) (that is, \(\Sigma =-\Sigma \)) 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.

阶次n的会议矩阵是一个\（n \ times n \）矩阵C，其对角线条目0和非对角线条目\（\ pm 1 \）满足\（CC ^ \ top =（n-1）I \）。如果C是对称的，则C具有对称谱\（\ Sigma \）（即\（\ Sigma =-\ Sigma \））和特征值\（\ pm \ sqrt {n-1} \）。我们证明了C的许多主要子矩阵也具有对称频谱，这导致具有对称频谱的图的Seidel矩阵（或等效地，完整带符号图的邻接矩阵）的示例。此外，我们证明了一些具有对称光谱的Seidel矩阵可以用这种构造来表征。