Mathematics of Control, Signals and Systems

, Volume 10, Issue 1, pp 31–40

The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate

  • John N. Tsitsiklis
  • Vincent D. Blondel

DOI: 10.1007/BF01219774

Cite this article as:
Tsitsiklis, J.N. & Blondel, V.D. Math. Control Signal Systems (1997) 10: 31. doi:10.1007/BF01219774


We analyze the computability and the complexity of various definitions of spectral radii for sets of matrices. We show that the joint and generalized spectral radii of two integer matrices are not approximable in polynomial time, and that two related quantities—the lower spectral radius and the largest Lyapunov exponent—are not algorithmically approximable.

Key words

Lyapunov exponent Lyapunov indicator Joint spectral radius Generalized spectral radius Discrete differential inclusion Computational complexity NP-hard Algorithmic solvability 

Copyright information

© Springer-Verlag London Limited 1997

Authors and Affiliations

  • John N. Tsitsiklis
    • 1
  • Vincent D. Blondel
    • 2
  1. 1.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute of MathematicsUniversity of LiégeLiègeBelgium

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