关键词: n维欧氏空间, n-1维超球面, 距离平方矩阵, 秩

Abstract:

Using the theory and method of distance geometry, we proved that a necessary and sufficient condition for Σ(A,N+1) being on an (n-1)dimensional hypersphere is that the matrix M(Σ(A,N+1))=(a²_kl)(k,l=0,1,…,N) has rank n+1, where M(Σ(A,N+1)) is the distance square matrix of Σ(A,N+1), Σ(A,N+1)={A₀,A₁,…,A_N} is an ndimensional finite set of points in ndimensional Euclidean space. A necessary and sufficient condition for 5 points on a 2dimensional hypersphere was given.

Key words: ndimensional Euclidean space, (n-1)dimensional , hypersphere, distance square matrix, rank