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Homework Help: Norm ineqaulity

  1. May 1, 2009 #1


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    1. The problem statement, all variables and given/known data

    Show that

    [tex]\frac{\Vert X(u+v) \Vert}{\Vert u+v \Vert} \leq \max \{
    \frac{\Vert Xu \Vert}{\Vert u \Vert}, \frac{\Vert Xv \Vert}{\Vert v \Vert} \}

    2. Relevant equations

    3. The attempt at a solution

    Tried to rewrite the max statement as an inequality (without loss of genreality). Then However I can't really get anyway with it since
    when I try to estimate the numerator or the denominator independently (triangle inequality, ...) I get a bound which is too high and I don't really know how to estimate both simultaniuously.

  2. jcsd
  3. May 2, 2009 #2
    this doesn't seem to be working. counter example:
    X=[[2,0],[0,1]] (x-coordinate is doubled)
    rescaled to the unit circle:
    applying the matrix X to these:
    but the lengths of the first two are both sqrt(2.5) < 2
  4. May 2, 2009 #3


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    Thanks for your answer. Now I'm slightly confused. Actually the example is taken from "Introduction to Applied Nonlinear Dynamical Systems". where it is stated that

    For any vectors [tex] f,g\in\mathbb{R}^n [/tex]
    [tex] \chi(f+g) \leq \max\{\chi(f),\chi(g)\} [/tex]

    where [tex]\chi[/tex] is the Lyapunov exponent given by.

    [tex] \chi(X,e) = \lim_{t\to\infty} \frac{1}{t} \log \frac{\vert Xe\vert}{\vert e \vert} [/tex]

    where X in general does depend on t.

    Since the logarithm is a montonuous function and i have to show the behavior for all [tex]t[/tex] such that it holds in the limit (or at least for some t>T). The book states that this follows readily from the defintion....

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