Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix

Hiroshi Kurata; Ravindra B. Bapat

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 270-282
  • ISSN: 2300-7451

Abstract

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By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.

How to cite

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Hiroshi Kurata, and Ravindra B. Bapat. "Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix." Special Matrices 4.1 (2016): 270-282. <http://eudml.org/doc/285882>.

@article{HiroshiKurata2016,
abstract = {By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.},
author = {Hiroshi Kurata, Ravindra B. Bapat},
journal = {Special Matrices},
keywords = {Euclidean distance matrix; Predistance matrix; Positive semidefinite matrix; hollow matrix; Moore-Penrose inverse; Laplacian matrix; Tree; predistance matrix; positive semidefinite matrix; tree},
language = {eng},
number = {1},
pages = {270-282},
title = {Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix},
url = {http://eudml.org/doc/285882},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Hiroshi Kurata
AU - Ravindra B. Bapat
TI - Moore-Penrose inverse of a hollow symmetric matrix and a predistance matrix
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 270
EP - 282
AB - By a hollow symmetric matrix we mean a symmetric matrix with zero diagonal elements. The notion contains those of predistance matrix and Euclidean distance matrix as its special cases. By a centered symmetric matrix we mean a symmetric matrix with zero row (and hence column) sums. There is a one-toone correspondence between the classes of hollow symmetric matrices and centered symmetric matrices, and thus with any hollow symmetric matrix D we may associate a centered symmetric matrix B, and vice versa. This correspondence extends a similar correspondence between Euclidean distance matrices and positive semidefinite matrices with zero row and column sums.We show that if B has rank r, then the corresponding D must have rank r, r + 1 or r + 2. We give a complete characterization of the three cases.We obtain formulas for the Moore-Penrose inverse D+ in terms of B+, extending formulas obtained in Kurata and Bapat (Linear Algebra and Its Applications, 2015). If D is the distance matrix of a weighted tree with the sum of the weights being zero, then B+ turns out to be the Laplacian of the tree, and the formula for D+ extends a well-known formula due to Graham and Lovász for the inverse of the distance matrix of a tree.
LA - eng
KW - Euclidean distance matrix; Predistance matrix; Positive semidefinite matrix; hollow matrix; Moore-Penrose inverse; Laplacian matrix; Tree; predistance matrix; positive semidefinite matrix; tree
UR - http://eudml.org/doc/285882
ER -

References

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  9. [9] C. R. Johnson and P. Tarazaga, Connections between the real positive semidefinite and distancematrices completion problems, Linear Algebra Appl., 223/224 (1995), 375-391.  Zbl0827.15032
  10. [10] H. Kurata and R. B. Bapat, Moore-Penrose inverse of a Euclidean distancematrix, Linear Algebra Appl., 472 (2015), 106-117.  Zbl1310.15060
  11. [11] I. J. Schoenberg, Remarks to Maurice Fréchet’s article “Sur la définition axiomatique d’une classe d’espace distanciés vectoriellement applicable sur l’espace de Hilbert” Ann. of Math. (2), 36, 1935, 724-732.  Zbl0012.30703
  12. [12] P. Tarazaga, T. L. Hayden and J. Wells, Circum-Euclidean distance matrices and faces, Linear Algebra Appl., 232 (1996), 77-96. [WoS] Zbl0837.15014

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