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2016 | 4 | 1 | 270-282
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
4
Numer
1
Strony
270-282
Opis fizyczny
Daty
otrzymano
2015-08-21
zaakceptowano
2016-06-10
online
2016-07-07
Twórcy
  • Graduate School of Arts and Sciences, The University of Tokyo, Tokyo, 153-8902, Japan
  • Indian Statistical Institute, New Delhi, 110016, India
Bibliografia
  • [1] R. Balaji and R. B. Bapat, On Euclidean distance matrices, Linear Algebra Appl., 424 (2007), 108-117.
  • [2] R. B. Bapat, Graphs and matrices (2nd ed.), Springer, 2014.
  • [3] R. B. Bapat, S. J. Kirkland and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl., 401 (2005), 193- 209.
  • [4] A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, Wiley-Interscience, 1974.
  • [5] F. Critchley, On certain linear mappings between inner products and squared distance matrices, Linear Algebra Appl., 105 (1988), 91-107.
  • [6] J. C. Gower, Properties of Euclidean and non-Euclidean distance matrices. Linear Algebra Appl., 67 (1985),81-97. [WoS]
  • [7] R. L. Graham and L. Lovász, Distance matrix polynomials of trees, Adv. Math., 29 (1978), no. 1, 60-88.
  • [8] R. L. Graham and H. O. Pollak, On the addressing problem for loop switching, Bell System Technology Journal, 50 (1971), 2495-2519.
  • [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.
  • [10] H. Kurata and R. B. Bapat, Moore-Penrose inverse of a Euclidean distancematrix, Linear Algebra Appl., 472 (2015), 106-117.
  • [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.
  • [12] P. Tarazaga, T. L. Hayden and J. Wells, Circum-Euclidean distance matrices and faces, Linear Algebra Appl., 232 (1996), 77-96. [WoS]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_spma-2016-0028
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