I was generating random matrices in Mathematica and noticed that the largest singular value $\sigma_\max$ is always greater than the largest entry of the matrix. I.e., for a matrix $A=\left(a_{ij}\right) \in \mathbb{R}^{m \times n}$ $$\sigma_\max \geq a_{ij}\quad \text{ for all }\;i,j$$
Unfortunantly I have no idea how to prove this fact (if it's true) and what does it mean. I'll apreciate any suggestions.
It is well known (and stated here, for example) that $$ \sigma_{max}(A) = \max_{\|x\| = 1} \|Ax\| $$ Thus: for any $i,j$: if $e_1,\dots,e_n$ denotes the canonical basis of $\Bbb R^n$, we have $$ \sigma_{max}(A) \geq \|A e_j\| = \sqrt{\sum_{i=1}^m a_{ij}^2} \geq \sqrt{a_{ij}^2} = |a_{ij}| $$ Thus, we get the desired inequality.