Let ${\bf A}$ be a $m \times n$ matrix with entries $a_{ij}$, and ${\bf x}$ be a $n \times 1$ vector with entries $x_{i}$. Then how can I show $$ \left\vert\,\sum_{j\ =\ 1}^{n} a_{1j}\,x_{j}\,\right\vert^{\, 2}\ \leq\ \sum_{j\ =\ 1}^{n}\left\vert\, a_{1j}\,\right\vert^{\, 2} \sum_{j\ =\ 1}^{n}\left\vert\, x_{j}\,\right\vert^{\, 2} $$
Thank you.
One of the proofs is the following that uses quadratic function which I like:
$0\leq \displaystyle \sum_{j=1}^n(a_{1j}-\lambda x_j)^2 = \displaystyle \sum_{j=1}^n a_{1j}^2 - 2\lambda\displaystyle \sum_{j=1}^n a_{1j}x_j + \lambda^2\displaystyle \sum_{j=1}^n x_j^2 = f(\lambda), \forall \lambda \in \mathbb{R} \Rightarrow \triangle' \leq 0 \Rightarrow \left(\displaystyle \sum_{j=1}^n a_{1j}x_j\right)^2 - \displaystyle \sum_{j=1}^n a_{1j}^2\displaystyle \sum_{j=1}^n x_j^2 \leq 0$. $\text{Q.E.D}$