Simple quadratic inequality

Question

Show that

$${\bf x^T V x}\leq \left( \sum_{i=1}^n v_{ii}x_i \right)^2$$

knowing that $\bf V$ is symmetric and positive semidefinite.

It should be simple, but can't figure it out. Thanks!

Answer 1

not true. Take $$ V = \left( \begin{array}{cc} \frac{1}{2} & \frac{1}{4} \\ \frac{1}{4} & \frac{1}{2} \\ \end{array} \right) $$

Note that no inequality in either direction can apply for all symmetric matrices $V$ and all vectors $x,$ as your displayed inequality is, on the left hand side, homogeneous degree one in the entries of $V,$ but degree two in the entries of $V$ on the right hand side.