I need to know the relation between operator norm of a matrix i.e. $ \Vert A\Vert_p$ for case of p=1 and 2 and its entry wise Frobenius norm $ \Vert A\Vert_F$.
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If $A$ is $n\times n$, then $$\frac1{\sqrt n}\|A\|_F\le\|A\|_1\le\sqrt n\,\|A\|_F,\qquad \|A\|_2\le\|A\|_F\le\sqrt n\,\|A\|_2.$$ More generally, if $A$ is $n\times m$, then $$\frac1{\sqrt m}\|A\|_F\le\|A\|_1\le\sqrt n\,\|A\|_F,\qquad \|A\|_2\le\|A\|_F\le\min(\sqrt n\,,\sqrt m)\,\|A\|_2.$$ To see that these inequalities are sharp, take respectively the matrices $$1_n\otimes \vec e^1,\quad \vec e^1\otimes 1_m,\qquad 1_n\otimes 1_m$$ for the first three inequalities. For the last one, complete the identity matrix by zeroes.
answered Jul 3, 2015 at 5:27
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$\begingroup$ Thanks for reply. If i say $ A$ is of size $n\times m (m>>n)$, so what will be the factor in the above equation $\sqrt{n}$ or $\sqrt{m}$. $\endgroup$– AstroCommented Jul 3, 2015 at 5:39