Methods for constructing distance matrices and the inverse eigenvalue problem Original Research Article

Let image and image be two distance matrices. We provide necessary conditions on image in order thatimagebe a distance matrix. We then show that it is always possible to border an n×n distance matrix, with certain scalar multiples of its Perron eigenvector, to construct an (n+1)×(n+1) distance matrix. We also give necessary and sufficient conditions for two principal distance matrix blocks D1 and D2 be used to form a distance matrix as above, where Z is a scalar multiple of a rank one matrix, formed from their Perron eigenvectors. Finally, we solve the inverse eigenvalue problem for distance matrices in certain special cases, including n=3,4,5,6, any n for which there exists a Hadamard matrix, and some other cases.