A - Papers appearing in refereed journals
Gower, J. C. 1985. Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra and its Applications. 67 (June), pp. 81-97. https://doi.org/10.1016/0024-3795(85)90187-9
Authors | Gower, J. C. |
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Abstract | A distance matrix D of order n is symmetric with elements , where dii=0. D is Euclidean when the quantities dij can be generated as the distances between a set of n points, X (n×p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p=rank(X) of any generating X; in general p+1 and p+2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when ρ=rank(D) it is shown that, depending on whether eTD−e is not or is zero, the generating points lie in either p=ρ−1 dimensions, in which case they lie on a hypersphere, or in p=ρ−2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p=r+s will comprise r real and s imaginary columns of X, and (r, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p+1=(r+1)+s or p+1=r+ (s+1) or p+2=(r+1)+(s+1), so that not only are r, s invariant, but they are both minimal for all admissible representations X. |
Year of Publication | 1985 |
Journal | Linear Algebra and its Applications |
Journal citation | 67 (June), pp. 81-97 |
Digital Object Identifier (DOI) | https://doi.org/10.1016/0024-3795(85)90187-9 |
Open access | Published as bronze (free) open access |
Publisher's version | Copyright license Publisher copyright |
Output status | Published |
Publication dates | |
01 Jun 1985 | |
Publication process dates | |
Accepted | 26 Jul 1984 |
Publisher | Elsevier Science Inc |
ISSN | 0024-3795 |
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