Generalized Euclidean distance matrices

Euclidean distance matrices ((Formula presented.)) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices ((Formula presented.) s) that include (Formula presented.) s. Each (Formula presented.) is an entry-wise nonnegative matrix. A (Formula presented.) is not symmetric unless it is an (Formula presented.). By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore–Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices.