# Norm Inequality: Proving Max Statement

• gop
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.

## Homework Statement

Show that

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

## 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

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

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 $$f,g\in\mathbb{R}^n$$
$$\chi(f+g) \leq \max\{\chi(f),\chi(g)\}$$

where $$\chi$$ is the Lyapunov exponent given by.

$$\chi(X,e) = \lim_{t\to\infty} \frac{1}{t} \log \frac{\vert Xe\vert}{\vert e \vert}$$

where X in general does depend on t.

Since the logarithm is a montonuous function and i have to show the behavior for all $$t$$ 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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