Equivalence of Norms Question

Question

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

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.