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Let $A$ be an $m\times n$ matrix such that $m < n$. I would like to know the conditions on $A$ such that the following is true:

$$\|Ax\| \leq \|Ay\| \implies \|x\| \leq \|y\|$$

It can easily be shown that if $\kappa(A)=1$ (condition number) then this property is satisfied. I am looking for the most general type of matrices that satisfy this condition.

Any help is much appreciated.

Thanks, Phanindra

  • 0
    Most likely the answer depends on the norms you use.2011-11-02
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    The matrix you are looking for simply does not exist. For any $m$-by-$n$ matrix $A$ (with $m), let $x$ be a nontrivial solution of $Ax=0$ and let $y=0$. Then $\|Ax\|=\|Ay\|=0$ but $\|x\|>\|y\|=0$, regardless of what norm is used.2011-11-02
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    @user1551: You are right. Thank you for the answer.2011-11-03

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