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Improving certain simple eigenvalue bounds

Published online by Cambridge University Press:  24 October 2008

Seppo Hyydö ,
Jorma Kaarlo Merikoski  and
Ari Virtanen
Seppo Hyydö
Affiliation:
University of Tampere, Finland
Jorma Kaarlo Merikoski
Affiliation:
University of Tampere, Finland
Ari Virtanen
Affiliation:
University of Tampere, Finland
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Extract

Throughout this paper we let A = (aij) be a non-zero n × n matrix-we study real matrices only–with row sums R1,…Rn and eigenvalues λ1,…,λn, ordered λ1≥…≥λn if they are real. We denote E = (1,…,1)T and su A = ΣiΣjaij = ETAE. If A is symmetric, a simple property of the Rayleigh quotient is that

satisfies

and

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 99 , Issue 3 , May 1986 , pp. 507 - 518
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1]Atkinson, F. V., Watterson, G. A. and Moran, P. A. P.. A. matrix inequality. Quart. J. Math. Oxford 11 (1960), 137–140.CrossRefGoogle Scholar
[2]Berman, A. and Plemmons, R. J.. Nonnegative Matrices in the Mathematical Sciences (Academic Press, 1979).Google Scholar
[3]Deutsch, E.. Lower bounds for the Perron root of a non-negative irreducible matrix. Math. Proc. Cambridge Philos. Soc. 92 (1982), 49–54.CrossRefGoogle Scholar
[4]Kankaanpää, H. and Merikoski, J. K.. Two inequalities for the sum of elements of a matrix. Linear and Multilinear Algebra 18 (1985), 9–22.CrossRefGoogle Scholar
[5]London, D.. Inequalities in quadratic forms. Duke Math. J. 33 (1966), 511–522.CrossRefGoogle Scholar
[6]London, D.. Two inequalities in nonnegative symmetric matrices. Pacific J. Math. 16 (1966), 515–536.Google Scholar
[7]Marcus, M. and Newman, M.. The sum of the elements of the powers of a matrix. Pacific J. Math. 12 (1962), 627–635.CrossRefGoogle Scholar
[8]Merikoshi, J. K.. On a lower bound for the Perron eigenvalue. BIT 19 (1979), 39–42.CrossRefGoogle Scholar
[9]Mirsky, L.. The spread of a matrix. Mathematika 3 (1956), 127–130.Google Scholar
[10]Mulholland, H. P. and Smith, C. A. B.. An inequality arising in genetical theory. Amer. Math. Monthly 66 (1959), 673–683.Google Scholar
[11]Parker, W. V.. Sets of complex numbers associated with a matrix. Duke Math. J. 15 (1948), 711–715.Google Scholar
[12]Wilkinson, J. H.. Householder's method for symmetric matrices. Numer. Math. 4 (1962), 354–361.Google Scholar
[13]Wolkowicz, H. and Styan, G. P. H.. Bounds for eigenvalues using traces. Linear Algebra Appl. 29 (1980), 471–506.Google Scholar