How to use the L1 norm constraint ||x||₁=1 in solving equations

[dirk_mec1]

Homework Statement

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Homework Equations

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The Attempt at a Solution

I think it is crucial that [itex]||x||_1=1[/tex] but I don't see how I can use that info. Can someone help me?[/itex]

 

[Pere Callahan]

Try to show
$$||A||_1 \geq \max_{1\leq j\leq n}{\sum_{i=1}^n{|a_{ij}|}}$$
and
$$||A||_1 \leq \max_{1\leq j\leq n}{\sum_{i=1}^n{|a_{ij}|}}$$
separately. Can you prover either of these statements?

Hint: For the first one it is enough to find an x with norm one such that
$$||Ax||_1 \geq \max_{1\leq j\leq n}{\sum_{i=1}^n{|a_{ij}|}}$$