Bounds of the matrix eigenvalues and its exponential by Lyapunov equation

Guang-Da Hu; Taketomo Mitsui

Kybernetika (2012)

  • Volume: 48, Issue: 5, page 865-878
  • ISSN: 0023-5954

Abstract

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We are concerned with bounds of the matrix eigenvalues and its exponential. Combining the Lyapunov equation with the weighted logarithmic matrix norm technique, four sequences are presented to locate eigenvalues of a matrix. Based on the relations between the real parts of the eigenvalues and the weighted logarithmic matrix norms, we derive both lower and upper bounds of the matrix exponential, which complement and improve the existing results in the literature. Some numerical examples are also given.

How to cite

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Hu, Guang-Da, and Mitsui, Taketomo. "Bounds of the matrix eigenvalues and its exponential by Lyapunov equation." Kybernetika 48.5 (2012): 865-878. <http://eudml.org/doc/251429>.

@article{Hu2012,
abstract = {We are concerned with bounds of the matrix eigenvalues and its exponential. Combining the Lyapunov equation with the weighted logarithmic matrix norm technique, four sequences are presented to locate eigenvalues of a matrix. Based on the relations between the real parts of the eigenvalues and the weighted logarithmic matrix norms, we derive both lower and upper bounds of the matrix exponential, which complement and improve the existing results in the literature. Some numerical examples are also given.},
author = {Hu, Guang-Da, Mitsui, Taketomo},
journal = {Kybernetika},
keywords = {Lyapunov equation; weighted logarithmic matrix norm; location of eigenvalues; bounds of the matrix exponential; Lyapunov equation; weighted logarithmic matrix norm; eigenvalues; matrix exponential; two-sided bounds},
language = {eng},
number = {5},
pages = {865-878},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Bounds of the matrix eigenvalues and its exponential by Lyapunov equation},
url = {http://eudml.org/doc/251429},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Hu, Guang-Da
AU - Mitsui, Taketomo
TI - Bounds of the matrix eigenvalues and its exponential by Lyapunov equation
JO - Kybernetika
PY - 2012
PB - Institute of Information Theory and Automation AS CR
VL - 48
IS - 5
SP - 865
EP - 878
AB - We are concerned with bounds of the matrix eigenvalues and its exponential. Combining the Lyapunov equation with the weighted logarithmic matrix norm technique, four sequences are presented to locate eigenvalues of a matrix. Based on the relations between the real parts of the eigenvalues and the weighted logarithmic matrix norms, we derive both lower and upper bounds of the matrix exponential, which complement and improve the existing results in the literature. Some numerical examples are also given.
LA - eng
KW - Lyapunov equation; weighted logarithmic matrix norm; location of eigenvalues; bounds of the matrix exponential; Lyapunov equation; weighted logarithmic matrix norm; eigenvalues; matrix exponential; two-sided bounds
UR - http://eudml.org/doc/251429
ER -

References

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  10. Hu, G. Da, Zhu, Q., Bounds of modulus of eigenvalues based on Stein equation., Kybernetika 46 (2010), 655-664. Zbl1205.15031MR2722093
  11. Kågström, B., 10.1007/BF01932398, BIT 17 (1977), 39-57. Zbl0356.65034MR0440896DOI10.1007/BF01932398
  12. Lancaster, P., Tismenetsky, M., The Theory of Matrices with Applications., Academic Press Inc. Orlando 1985. MR0792300
  13. Pao, C. V., 10.1016/0024-3795(73)90015-3, Linear Algebra Appl. 7 (1973), 159-164. Zbl0257.15016MR0320037DOI10.1016/0024-3795(73)90015-3
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  15. Ström, T., 10.1137/0712055, SIAM J. Numer. Anal. 12 (1975), 741-753. Zbl0321.15012MR0408227DOI10.1137/0712055

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