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I have a matrix where the $L_1$ norm on an row is equal to zero. My question is what can I say about the $L_2$ norm on any row of that matrix? Numerically with the example I have, computing the $L_2$ norm on each row is equivalent for each row - but I'm having trouble writing/finding a rule that substantiates this.

Thank you

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closed as unclear what you're asking by barto, MagicMan, quid, N. F. Taussig, kjetil b halvorsen Apr 12 at 15:55

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

Welcome to Math.Stackexchange. People will be able to help you better if you provide more context, attempts you have made at a solution etc. For your particular problem, could you show us the actual matrix you are working with? –  Galois Group Aug 31 '12 at 21:11

1 Answer 1

If $n\leq m$ for $n,m\in\{1,2,\dots\}$ you can say that $$\sqrt[n]{\sum_{i=1}^{k}x_{i}^{n}} \leq \sqrt[m]{\sum_{i=1}^{k}x_{i}^{m}}.$$

So for the same row, you can compare the $L_{1}$ and $L_{2}$ norms in this way. But the relationship of the norms of one row to the norms of another row depends on how (and whether) those rows are related.

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