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Maximum Norm Proof

Amer

Active member
Mar 1, 2012
275
Prove that for

$A \in \mathbb{R}^{n\times n} $
[tex]||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} | [/tex]

I know that
$||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $

such that $x \in \mathbb{R}^n$

any hints
 

dwsmith

Well-known member
Feb 1, 2012
1,673
Prove that for

$A \in \mathbb{R}^{n\times n} $
[tex]||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} | [/tex]

I know that
$||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $

such that $x \in \mathbb{R}^n$

any hints
I believe you can start with $||A||_p$ the p norm and take the limit as $p\to\infty$ to prove the problem.
Isn't the infinity norm just the max of $|a_i|$ not the sum of them?
 

Amer

Active member
Mar 1, 2012
275
I believe you can start with $||A||_p$ the p norm and take the limit as $p\to\infty$ to prove the problem.
Isn't the infinity norm just the max of $|a_i|$ not the sum of them?
for a vector it is the max of $|a_i|$
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,707
Prove that for

$A \in \mathbb{R}^{n\times n} $
[tex]||A||_{\infty} = \text{max}_{i=1,...,n} \sum_{j=1}^n |a_{ij} | [/tex]

I know that
$||A||_{\infty} = \text{max} \dfrac{||Ax||_{\infty} }{||x||_{\infty}} $

such that $x \in \mathbb{R}^n$

any hints
You might find it easier to use the equivalent definition $\|A\|_\infty = \max \{\|Ax\|_\infty : \|x\|_\infty \leqslant 1\}.$ For $\|x\|_\infty \leqslant 1$, show that $$\|Ax\|_\infty = \max_{1\leqslant i\leqslant n}|(Ax)_i| = \max_{1\leqslant i\leqslant n}\Bigl| \sum_{j=1}^n a_{ij}x_j \Bigr| \leqslant \sum_{j=1}^n |a_{ij} |.$$
For the reverse inequality, find $\|Ax\|$ where $x$ is the vector with a 1 in the $i$th coordinate and 0 for every other coordinate.