Methods for constructing distance matrices and the inverse eigenvalue problem

Let D-1 is an element of R-kxk and D-2 is an element of R-lxl be two distan ce matrices. We provide necessary conditions on Z is an element of R-kxl in order that [GRAPHICS] be a distance matrix. We then show that it is always possible to border an n x n distance matrix, with certain scalar multiples of its Perron eigenvec tor, to construct an (n + 1) x (n + 1) distance matrix. We also give necess ary and sufficient conditions for two principal distance matrix blocks Di a nd D2 be used to form a distance matrix as above, where Z is a scalar multi ple of a rank one matrix, formed from their Perron eigenvectors. Finally, w e solve the inverse eigenvalue problem for distance matrices in certain spe cial cases, including n = 3, 4, 5, 6, any n for which there exists a Hadama rd matrix, and some other cases. (C) 1999 Elsevier Science Inc. All rights reserved.