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RELATIVE PERTURBATION BOUNDS FOR THE JOINT SPECTRUM OF COMMUTING TUPLES OF MATRICES

Part of: Basic linear algebra

Published online by Cambridge University Press:  05 July 2018

ARNAB PATRA Open the ORCID record for ARNAB PATRA [Opens in a new window]  and
P. D. SRIVASTAVA Open the ORCID record for P. D. SRIVASTAVA [Opens in a new window]
ARNAB PATRA*
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India email arnptr91@gmail.com
P. D. SRIVASTAVA
Affiliation:
Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India email pds@maths.iitkgp.ernet.in
*
✉arnptr91@gmail.com
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Abstract

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In this paper, we study the relative perturbation bounds for joint eigenvalues of commuting tuples of normal $n\times n$ matrices. Some Hoffman–Wielandt-type relative perturbation bounds are proved using the Clifford algebra technique. We also extend a result for diagonalisable matrices which improves a relative perturbation bound for single matrices.

Keywords

joint eigenvaluesClifford algebracommuting tuple of matrices

MSC classification

Primary: 15A42: Inequalities involving eigenvalues and eigenvectors
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 3 , December 2018 , pp. 414 - 421
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

References

Bhatia, R. and Bhattacharyya, T., ‘A generalization of the Hoffman–Wielandt theorem’, Linear Algebra Appl. 179 (1993), 11–17.Google Scholar
Eisenstat, S. C. and Ipsen, I. C. F., ‘Three absolute perturbation bounds for matrix eigenvalues imply relative bounds’, SIAM J. Matrix Anal. Appl. 20(1) (1998), 149–158.Google Scholar
Freedman, D. Z., ‘Perturbation bounds for the joint spectrum of commuting matrices’, Linear Algebra Appl. 387 (2004), 29–40.Google Scholar
Hoffman, A. J. and Wielandt, H. W., ‘The variation of the spectrum of a normal matrix’, Duke Math. J. 20(1) (1953), 37–39.Google Scholar
Li, W. and Chen, J.-X., ‘The eigenvalue perturbation bound for arbitrary matrices’, J. Comput. Math. 24 (2006), 141–148.Google Scholar
Li, W. and Sun, W., ‘The perturbation bounds for eigenvalues of normal matrices’, Numer. Linear Algebra Appl. 12(2–3) (2005), 89–94.Google Scholar
McIntosh, A. and Pryde, A., ‘A functional calculus for several commuting operators’, Indiana Univ. Math. J. 36(2) (1987), 421–439.Google Scholar
Pryde, A. J., ‘Inequalities for the joint spectrum of simultaneous triangularizable matrices’, in: Miniconf. Probability and Analysis (Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra, 1992), 196–207.Google Scholar
Stewart, G. W. and Sun, J. G., Matrix Perturbation Theory (Academic Press, New York, 1990).Google Scholar
Sun, J. G., ‘On the variation of the spectrum of a normal matrix’, Linear Algebra Appl. 246 (1996), 215–223.Google Scholar