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$.



Comparison of Lp norm of matrix and its entry wise norm.
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.


