I have a matrix $\textbf{A}\in\mathbb{C}^{n\times n}$ with singular values $\left\{\sigma_i\right\}_{i=1}^n$. How do I show that $$\sigma_1>\max_{i,j}|\textbf{A}_{ij}|,$$ where $\sigma_1$ is the maximum singular value?
Here is my attempt. Starting with the Frobenius norm $\|\textbf{A}\|_F$, \begin{align} \|\textbf{A}\|_F&=\sqrt{\sum_{j=1}^n\sum_{i=1}^n|\textbf{A}_{ij}|^2}\\ &=\sqrt{\text{tr}\left\{\textbf{A}^H\textbf{A}\right\}}\\ &=\sqrt{\sum_{i=1}^n\sigma_i^2}\\ &\le\sqrt{n\sigma_1^2} \end{align} and so $\sqrt{n\sigma_1^2}\ge\sqrt{\sum_{j=1}^n\sum_{i=1}^n|\textbf{A}_{ij}|^2}$.
How can I proceed?