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Singular Values

In: Lyapunov Exponents

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  • Luís Barreira

    (Universidade de Lisboa, Instituto Superior Técnico)

Abstract

In this chapter we go back to the core of the theory of Lyapunov exponents and the theory of regularity. It has the main purpose of describing in detail the relation between singular values and Lyapunov exponents, both for discrete and continuous time. We first show that the general inequalities between the values of the Lyapunov exponent and of the upper exponential growth rates of the singular values are the best possible. More precisely, we show that any sets of numbers satisfying these general inequalities are realized as the values of the Lyapunov exponent and of the upper exponential growth rates of the singular values of some bounded sequence of matrices. We then establish one of the central results of the book: for an arbitrary tempered sequence of matrices, and so possibly nonregular, we obtain general inequalities between the values of the Lyapunov exponent and the lower and upper exponential growth rates of the singular values (and not only upper). These inequalities are obtained as a consequence of the existence of a structure of Oseledets type that is present even for a nonregular sequence. We also consider briefly the case of continuous time and we obtain corresponding versions of the results.

Suggested Citation

  • Luís Barreira, 2017. "Singular Values," Springer Books, in: Lyapunov Exponents, chapter 0, pages 115-135, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-71261-1_6
    DOI: 10.1007/978-3-319-71261-1_6
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