A Complete Characterization of Linear Dependence and Independence for All 4-by-4 Metric Matrices

In this article, we study the properties of 4-by-4 metric matrices and characterize their dependence and independence by M4×4=(M4×4−DM4×4)∪DM4×4, where DM4×4 is the set of all dependent metric matrices. DM4×4 is further characterized by DM4×4=DM14×4∪DM24×4, where DM24×4 is characterized by DM24×4=DM214×4∪DM224×4. These characterizations provide some insightful findings that go beyond the Euclidean distance or Euclidean distance matrix and link the distance functions to vector spaces, which offers some theoretical and application-related advantages. In the application parts, we show that the metric matrices associated with all Minkowski distance functions over four different points are linearly independent, and that the metric matrices associated with any four concyclic points are also linearly independent.