Norm Inequality: Proving Max Statement

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In summary, the conversation discusses trying to show that a certain mathematical statement holds true, but the attempt at a solution using an example does not seem to work. The conversation also mentions a related topic from a book, but it is unclear how it relates to the original problem.
  • #1

gop

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Homework Statement



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]

Homework Equations




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
 
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  • #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
 
  • #3
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
 

1. What is norm inequality?

Norm inequality is a mathematical concept that compares the size of two vectors or matrices in a vector space. It states that the norm (or magnitude) of a sum or difference of two vectors is less than or equal to the sum of their individual norms.

2. How is norm inequality used in proving max statements?

Norm inequality is used in proving max statements by providing a way to compare the magnitude of different quantities. It allows us to simplify complex expressions and make them easier to analyze, which is crucial in proving max statements.

3. Can norm inequality be applied to any type of vector or matrix?

Yes, norm inequality can be applied to any type of vector or matrix in a vector space. It is a fundamental concept in linear algebra and is used in various fields of mathematics and science.

4. What are the key steps in proving a max statement using norm inequality?

The key steps in proving a max statement using norm inequality include defining the problem, identifying the quantities to be compared, applying the norm inequality, simplifying the expression, and arriving at the desired conclusion.

5. Are there any limitations to using norm inequality in proving max statements?

While norm inequality is a powerful tool in proving max statements, it does have some limitations. It may not be applicable in cases where the vectors or matrices involved do not have well-defined norms, or in cases where the underlying vector space is not a normed space.

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