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A Relation Between the Weighted Logarithmic Norm of a Matrix and the Lyapunov Equation

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Abstract

In this note, we define a weighted logarithmic norm for any matrix. In the case when a stable matrix A is considered, we obtain the relationship between the maximal eigenvalue of a symmetric positive definite matrix H which is a solution of the Lyapunov equation and the weight H logarithmic norm of A. It can be seen that the weighted logarithmic norm of A is always a negative value in this case. Several examples illustrate the relationship.

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REFERENCES

  1. A. Bellen, N. Guglielmi, and A. E. Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Trans. Circuits Systems I Fund. Theory Appl., 46:1 (1999), pp. 212–216.

    Google Scholar 

  2. R. A. DeCarlo, Linear Systems, Prentice-Hall, Englewood Cliffs, NJ, 1989.

    Google Scholar 

  3. K. Dekker and J. G. Verwer, Stability of Runge–Kutta methods for stiff nonlinear differential equations, North-Holland, Amsterdam, 1984.

  4. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input–Output Properties, Academic Press, New York, 1975.

    Google Scholar 

  5. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.

    Google Scholar 

  6. G. Di Hu and G. Da Hu, Some simple criteria for stability of neutral delaydifferential systems, Appl. Math. Comput., 80(1996), pp. 257–271.

    Google Scholar 

  7. P. Lancaster and M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, FL, 1985.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Memorial University of Newfoundland, St. John's, NF, Canada, A1B 3X5.

    Guang-Da Hu

  2. Systems Control Group, Department of Electrical and Computer Engineering, University of Toronto, 10 King's College Road, Toronto, Ontario, Canada, M5S 3G4

    Guang-Di Hu

Authors
  1. Guang-Da Hu
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  2. Guang-Di Hu
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Hu, GD., Hu, GD. A Relation Between the Weighted Logarithmic Norm of a Matrix and the Lyapunov Equation. BIT Numerical Mathematics 40, 606–610 (2000). https://doi.org/10.1023/A:1022380116022

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  • DOI: https://doi.org/10.1023/A:1022380116022

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