Encyclopedia of Social Network Analysis and Mining

2018 Edition
| Editors: Reda Alhajj, Jon Rokne

Iterative Methods for Eigenvalues/Eigenvectors

Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-7131-2_148

Synonyms

Glossary

Eigenvalue/Eigenvector

The fundamental entities that characterize any given matrix and can be obtained by finding the roots of the characteristic polynomial of the matrix or by iterative methods

Social Network Analysis

A research area in social and behavioral sciences that uses networks to represent and hence analyze social phenomena

Iterative Method

A procedure for solving a problem by generating a sequence of improving approximations to the true solution of the given problem

Definition

Eigenvalues and eigenvectors are fundamental concepts in linear algebra (Golub and Van Loan 2012; Golub and Vorst 2000) and are defined as follows:

Definition 1

Let A be an n-by-n real matrix (i.e., in \( {\mathbb{R}}^{n\times n} \)
This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

The work of the second author was supported in part by the Natural Science Foundation of Zhejiang Province and National Natural Science Foundation of China (Grant Nos. Y6110639, 11201422).

References

  1. Arnoldi WE (1951) The principle of minimized iteration in the solution of the matrix eigenproblem. Q Appl Math 9:17–29, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar
  2. Francis J (1961) The QR transformation: a unitary analogue to the LR transformation, parts I and II. Comput J 4:265–272, 332–345, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar
  3. Golub GH, Van Loan CF (2012) Matrix computations, 4th edn. Johns Hopkins University Press, BaltimoreMATHGoogle Scholar
  4. Golub GH, Vorst H (2000) Eigenvalue computation in the 20th century. J Comput Appl Math 123:35–65, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar
  5. Horn RA, Johnson CR (1985) Matrix analysis. Cambridge University Press, London, MATHMATHCrossRefGoogle Scholar
  6. Householder AS (1958) Unitary triangularization of a nonsymmetric matrix. J ACM 5:339–342, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar
  7. Householder AS (1964) The theory of matrices in numerical analysis. Dover, New York, MATHMATHGoogle Scholar
  8. Kamvar S, Haveliwala T, Golub GH (2004) Adaptive methods for the computation of PageRank. Linear Algebra Appl 386:51–65, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar
  9. Katz L (1953) A new status index derived from sociometric analysis. Psychometrika 18:39–43, MATHMATHCrossRefGoogle Scholar
  10. Newman MEJ (2009) Networks: an introduction. Oxford University Press, OxfordGoogle Scholar
  11. Opsahl T, Agneessens F, Skvoretz J (2010) Node centrality in weighted networks: generalizing degree and shortest paths. Soc Network 32(3):245–251CrossRefGoogle Scholar
  12. Page L, Brin S, Motwani R, Winograd T (1998) The PageRank citation ranking: bringing order to the web. Stanford Digital Libraries Working PaperGoogle Scholar
  13. Parlett BN (1980) The symmetric eigenvalue problem. Prentice-Hall, Englewood Cliffs, MATHMATHGoogle Scholar
  14. Saad Y (2003) Iterative methods for sparse linear systems, 2nd edn. SIAM, Philadelphia, MATHMATHCrossRefGoogle Scholar
  15. Schur I (1909) On the characteristic roots of a linear substitution with an application to the theory of integral equations. Math Ann 66:488–510, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar
  16. Stewart GW (1973) Introduction to matrix computations. Academic, New York, MATHMATHGoogle Scholar
  17. Wilkinson JH (1965) The algebraic eigenvalue problem. Clarendon, Oxford, MATHMATHGoogle Scholar
  18. Yin J, Yin G, Ng M (2012) On adaptively accelerated Arnoldi method for computing PageRank. Numer Linear Algebra Appl 19(1):73–85, MATHMathSciNetMathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsThe Chinese University of Hong KongShatin, NT Hong Kong SARChina
  2. 2.College of Statistics and MathematicsZhejiang Gongshang UniversityHangzhouChina

Section editors and affiliations

  • Suheil Khoury
    • 1
  1. 1.American University of SharjahSharjahUnited Arab Emirates