Kybernetika 48 no. 5, 865-878, 2012

Bounds of the matrix eigenvalues and its exponential by Lyapunov equation

Guang-Da Hu and Taketomo Mitsui


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.


Lyapunov equation, weighted logarithmic matrix norm, location of eigenvalues, bounds of the matrix exponential


15A18, 15A60, 34D20


  1. D. S. Bernstein: Matrix Mathematics. Princeton University Press, Princeton and Oxford 2005.   CrossRef
  2. K. Dekker and J. G. Verwer: Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. North-Holland, Amsterdam 1984.   CrossRef
  3. C. A. Desoer and M. Vidyasagar: Feedback Systems: Input-output Properties. Academic Press, New York 1975.   CrossRef
  4. G. H. Golub and C. F. Van Loan: Matrix Computations. Third edition. Johns Hopkins University Press, Baltimore 1996.   CrossRef
  5. R. A. Horn and C. R. Johnson: Matrix Analysis. Cambridge University Press, Cambridge 1985.   CrossRef
  6. R. A. Horn and C. R. Johnson: Topics in Matrix Analysis. Cambridge University Press, Cambridge 1991.   CrossRef
  7. G. Da Hu and G. Di Hu: A relation between the weighted logarithmic norm of matrix and Lyapunov equation. BIT 40 (2000), 506-510.   CrossRef
  8. G. Da Hu and M. Z. Liu: The weighted logarithmic matrix norm and bounds of the matrix exponential. Linear Algebra Appl. 390 (2004), 145-154.   CrossRef
  9. G. Da Hu and M. Z. Liu: Properties of the weighted logarithmic matrix norms. IMA. J. Math. Control Inform. 25 (2008), 75-84.   CrossRef
  10. G. Da Hu and Q. Zhu: Bounds of modulus of eigenvalues based on Stein equation. Kybernetika 46 (2010), 655-664.   CrossRef
  11. B. K{\aa}gström: Bounds and perturbation bounds for the matrix exponential. BIT 17 (1977), 39-57.   CrossRef
  12. P. Lancaster and M. Tismenetsky: The Theory of Matrices with Applications. Academic Press Inc. Orlando 1985.   CrossRef
  13. C. V. Pao: Logarithmic derivatives of a square matrix. Linear Algebra Appl. 7 (1973), 159-164.   CrossRef
  14. W. J. Rugh: Linear System Theory. Prentice Hall, Upper Saddle River, New Jersey 1996.   CrossRef
  15. T. Ström: On logarithmic norms. SIAM J. Numer. Anal. 12 (1975), 741-753.   CrossRef