Equivalence of vector norms

Nov 2009
234
12
Norway
Hi.

problem:
For each of the following, verify the inequality and give an example of a nonzero vector or matrix for which equality is achieved. In this problem x is an m-vector and A is an \(\displaystyle m \times n\) matrix.

(a) \(\displaystyle ||x||_{\infty} \leq ||x||_2\),

(b) \(\displaystyle ||x||_2 \leq \sqrt{m} ||x||_{\infty}\),

(c) \(\displaystyle ||A||_{\infty} \leq \sqrt{n} ||A||_2\),

(d) \(\displaystyle ||A||_2 \leq \sqrt{m} ||A||_{\infty}\).

attempt:

(a) I rewrite the inequality so it is easier for me to read:
\(\displaystyle (max_i |x_i|)^2 \leq (x^2_1+\cdots +x^2_m)\)
Let \(\displaystyle x_k\) be the largest component of \(\displaystyle x\). Then \(\displaystyle (max_i|x_i|)^2=x^2_k\).
If the remaining components are zero, the inequality becomes an equality. If one or more of the remaining components are nonzero, it becomes an inequality.
Example of vector: \(\displaystyle x=[x_k,0,\cdots,0]\).

I'm sure there are better ways to verify this.

(b) Again, I rewrite:

\(\displaystyle x^2_1+\cdots+x^2_m \leq m(max_i|x_i|)^2\).
Let \(\displaystyle x_k\) be the largest component of \(\displaystyle x\).
Since \(\displaystyle mx^2_k=x^2_k+\cdots +x^2_k\) (m-times) and since \(\displaystyle x_i\) for\(\displaystyle i\neq k\) are smaller than \(\displaystyle x_k\), the inequality holds.
It's an equality for all vectors \(\displaystyle x\) with \(\displaystyle m=1\) :)

I am not at all sure what to do with (c) and (d).
Any hints are appreciated.
 
Feb 2010
422
141
Is \(\displaystyle ||A||_p\) the regular p norm when A is regarded as a vector in \(\displaystyle \mathbb{R}^{nm}\)?
 
Nov 2009
234
12
Norway
Is \(\displaystyle ||A||_p\) the regular p norm when A is regarded as a vector in \(\displaystyle \mathbb{R}^{nm}\)?
\(\displaystyle ||A||_p=\left( \sum^m_{i=1} \sum^n_{j=1} |a_{ij}|^p \right)^{1/p}\)

Is this what you mean?
 
Feb 2010
422
141
Yes. A matrix is a block of nm numbers, so it can be identified with a vector in R^{nm}, and its norm is the same as the norm of this vector, so this makes (a) stronger than (c) I believe. For (d), consider an n by 1 matrix all of whose entries are C>0. Then (d) asserts \(\displaystyle nmC^2 \leq mC^2\) which is false if n>1, so perhaps there was a typo? If m is replaced by nm then (d) follows from (b) in the same way that (c) follows from (a).
 
Nov 2009
234
12
Norway
If I look at the matrix A as a \(\displaystyle mn\)-dimensional vector and then calculate the, say 2-norm, I am calculating the Forbenius norm. \(\displaystyle ||A||_2\) is the matrix norm induced by vector norms, and so is not the same as the Forbenius norm.
Am I way off here?