1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Norm ineqaulity

  1. May 1, 2009 #1

    gop

    User Avatar

    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} \}
    [/tex]

    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.

    thx
     
  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)
    u=(1,1)
    v=(1,-1)
    u+v=(2,0)
    rescaled to the unit circle:
    u/|u|=(1/sqrt(2),1/sqrt(2))
    v/|v|=(1/sqrt(2),-1/sqrt(2))
    u+v/|u+v|=(1,0)
    applying the matrix X to these:
    X(u/|u|)=(2/sqrt(2),1/sqrt(2))
    X(v/|u|)=(2/sqrt(2),-1/sqrt(2))
    X(u+v/|u+v|)=(2,0)
    but the lengths of the first two are both sqrt(2.5) < 2
     
  4. May 2, 2009 #3

    gop

    User Avatar

    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....

    thx
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Norm ineqaulity
  1. Is this a norm? (Replies: 13)

  2. Gaussian Norms (Replies: 1)

  3. Norm of a Matrix (Replies: 4)

Loading...