Proving a simple vector inequality

Question

Can you prove that, if $\vec{x}, \vec{y}$ are real vectors and $\vec{x}$'s elements are nonnegative, then $$ \sum_i x_i \sum_j x_j y_j^2 \geq \left( \sum_i x_i y_i \right)^2 $$ I thought it followed from Cauchy-Shwartz but that does not seem to be sufficient.

Answer 1

Use Cauchy-Schwartz with $a_i = x_i^{1/2}$ and $b_i = x_i^{1/2}y_i$.

\begin{align} \left(\sum_i a_i b_i \right)^2 &\le \left(\sum_i a_i^2 \right) \left(\sum_i b_i^2 \right) \\ \left(\sum_i x_i y_i \right)^2 &\le \left(\sum_i x_i \right) \left(\sum_i x_i y_i^2 \right) \end{align}