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

• Autorzy: Hiroshi Kurata , Ravindra B. Bapat
• 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.

Słowa kluczowe

EN

Euclidean distance matrix; Predistance matrix; Positive semidefinite matrix; hollow matrix; Moore-Penrose inverse; Laplacian matrix; Tree

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2016
• Tom: 4
• Numer: 1
• Strony: 270-282

Daty

otrzymano

2015-08-21

zaakceptowano

2016-06-10

online

2016-07-07

Twórcy

autor

Hiroshi Kurata

• Graduate School of Arts and Sciences, The University of Tokyo, Tokyo, 153-8902, Japan

autor

Ravindra B. Bapat

• Indian Statistical Institute, New Delhi, 110016, India

Bibliografia

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• [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]

Identyfikatory

DOI

10.1515/spma-2016-0028