Geodesic
Distance matrices for points on a~line, on~a~circle, and at the vertices of an $n$-dimensional cube
Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 127-138.

Voir la notice de l'article provenant de la source Math-Net.Ru

For $n$ points $A_i$, $i=1,2,\dots,n$, in Euclidean space $\mathbb R^m$, the distance matrix is defined as a matrix of the form $D=(D_{i,j})_{\substack{i=1,n\\j=1,n}}$, where the $D_{i,j}$ are the distances between the points $A_i$ and $A_j$ . Two configurations of points $A_i$, $i=1,2,\dots,n$, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of an $m$-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix. 
@article{MZM_1995_58_1_a10,
     author = {S. M. \`Ertel'},
     title = {Distance matrices for points on a~line, on~a~circle, and at the vertices of an $n$-dimensional cube},
     journal = {Matemati\v{c}eskie zametki},
     pages = {127--138},
     publisher = {mathdoc},
     volume = {58},
     number = {1},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a10/}
}
TY  - JOUR
AU  - S. M. Èrtel'
TI  - Distance matrices for points on a~line, on~a~circle, and at the vertices of an $n$-dimensional cube
JO  - Matematičeskie zametki
PY  - 1995
SP  - 127
EP  - 138
VL  - 58
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a10/
LA  - ru
ID  - MZM_1995_58_1_a10
ER  - 
%0 Journal Article
%A S. M. Èrtel'
%T Distance matrices for points on a~line, on~a~circle, and at the vertices of an $n$-dimensional cube
%J Matematičeskie zametki
%D 1995
%P 127-138
%V 58
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a10/
%G ru
%F MZM_1995_58_1_a10
S. M. Èrtel'. Distance matrices for points on a~line, on~a~circle, and at the vertices of an $n$-dimensional cube. Matematičeskie zametki, Tome 58 (1995) no. 1, pp. 127-138. http://geodesic.mathdoc.fr/item/MZM_1995_58_1_a10/

[1] Babaliev A. M., “Ob odnom metode interpolirovaniya funktsii mnogikh nezavisimykh peremennykh”, Mashinnaya grafika i ee primenenie, VTs SO AN SSSR, Novosibirsk, 1973

[2] Micchelli C. A., “Interpolation of scattered data: Distance matrices and conditionally positive definite functions”, Constr. Approx., 2 (1986), 11–22 | DOI | MR | Zbl

[3] Marcus M., Smith T. R., “A note on the derminants and eigenvalues of distance matrices”, Linear and Multilinear Algebra, 25 (1989), 219–230 | DOI | MR | Zbl

[4] Adamar Zh., Elementarnaya geometriya. Ch. 2. Stereometriya, M., 1938