The Effect of a Transpose on a Matrix Inequality

Question

In the solution to an exercise I came across the following: $y^TA_N \geq c_N^T \rightarrow A_N^Ty \geq c_N$. Now I was wondering, is it in general true that an inequality remains valid when 'taking transposes on both sides'? If so, what is the proof for this?

Answer 1

Generally one says $A\ge B$, for matrices, if each entry of $A$ is $\ge$ each entry of $B$. This is true if and only if each entry of $A^T$ is $\ge$ each entry of $B^T$, i.e. $A^T\ge B^T$.