The matrix is an Euclidean distance (EDM) if there exist points so that the matrix elements are squares of the euclidean distances between these points.
In this work, we prove some important properties of EDM. Then we focus on the inverse eigenvalue problem for EDM.
The inverse eigenvalue problem is as follows: to construct (or to prove the existence of) a matrix with a given spectrum and required properties (in particular that the matrix is EDM).
It is well known that the IEP for EDM of size 3 has a solution. Here we find all the solutions to this problem, we study their connection with geometry and possible extension to larger EDM using bordered matrices.
Then we show the connection between the well known problem of the existence of Hadamard matrices and the IEP for EDM.