Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-24T11:06:10.034Z Has data issue: false hasContentIssue false
  • English
  • Français

INVERSE EIGENVALUE PROBLEM FOR EUCLIDEAN DISTANCE MATRICES OF SIZE 3

Part of: Basic linear algebra

Published online by Cambridge University Press:  15 October 2012

GAŠPER JAKLIČ  and
JOLANDA MODIC
GAŠPER JAKLIČ
Affiliation:
FMF and IMFM, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia IAM, University of Primorska, Jadranska 19, 1000 Ljubljana, Slovenia (email: [email protected])
JOLANDA MODIC*
Affiliation:
FMF, University of Ljubljana and XLAB d.o.o., Pot za Brdom 100, 1000 Ljubljana, Slovenia (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A matrix is a Euclidean distance matrix (EDM) if there exist points such that the matrix elements are squares of distances between the corresponding points. The inverse eigenvalue problem (IEP) is as follows: construct (or prove the existence of) a matrix with particular properties and a given spectrum. It is well known that the IEP for EDMs of size 3 has a solution. In this paper all solutions of the problem are given and their relation with geometry is studied. A possible extension to larger EDMs is tackled.

Keywords

Euclidean distance matrixinverse eigenvalue problembordered matrix

MSC classification

Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 82 - 93
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

This research was funded in part by the European Union, European Social Fund, Operational Programme for Human Resources, Development for the Period 2007–2013.

References

[1]Balaji, R. and Bapat, R. B., ‘Block distance matrices’, Electron. J. Linear Algebra 16 (2007), 435–443.CrossRefGoogle Scholar
[2]Chu, M. T., ‘Constructing a Hermitian matrix from its diagonal entries and eigenvalues’, SIAM J. Matrix Anal. Appl. 16 (1995), 207–217.CrossRefGoogle Scholar
[3]Chu, M. T. and Golub, G. H., Inverse Eigenvalue Problems: Theory, Algorithms, and Applications (Oxford University Press, Oxford, 2005).CrossRefGoogle Scholar
[4]Dattorro, J., Convex Optimization and Euclidean Distance Geometry (Meboo, Palo Alto, CA, 2005).Google Scholar
[5]Fiedler, M., ‘Eigenvalues of nonnegative symmetric matrices’, Linear Algebra Appl. 9 (1974), 119–142.CrossRefGoogle Scholar
[6]Fiedler, M., ‘Some inverse problems for elliptic matrices with zero diagonal’, Linear Algebra Appl. 332–334 (2001), 197–204.CrossRefGoogle Scholar
[7]Golub, G. H. and van Loan, C. F., Matrix Computations, 3rd edn (Johns Hopkins University Press, Baltimore, MD, 1996).Google Scholar
[8]Hayden, T. L., Reams, R. and Wells, J., ‘Methods for constructing distance matrices and the inverse eigenvalue problem’, Linear Algebra Appl. 295 (1999), 97–112.CrossRefGoogle Scholar
[9]Jaklič, G. and Modic, J., ‘On properties of cell matrices’, Appl. Math. Comput. 216 (2010), 2016–2023.Google Scholar
[10]Laffey, T. J. and Šmigoc, H., ‘Construction of nonnegative symmetric matrices with given spectrum’, Linear Algebra Appl. 421 (2007), 97–109.CrossRefGoogle Scholar
[11]Schoenberg, I. J., ‘Metric spaces and positive definite functions’, Trans. Amer. Math. Soc. 44 (1938), 522–536.CrossRefGoogle Scholar