Is this quantity positive?

Question

$$\langle {A\mathbf{x}_1, \mathbf{x}_1} \rangle (\frac{\lambda}{2} -\frac{\lambda^2}{2} ) + \langle {A\mathbf{x}_2, \mathbf{x}_2} \rangle (\frac{1 - \lambda}{2} -\frac{(1-\lambda)^2}{2} ) - \frac{\lambda(1-\lambda)}{2} (\langle {A\mathbf{x}_1, \mathbf{x}_2} \rangle + \langle {A\mathbf{x}_2, \mathbf{x}_1} \rangle) $$

for $\lambda \in (0,1)$. $A \in \mathcal{M}_{n,n} (\mathbb{R})$ being symmetric positive-definite.

Answer 1

Yes it is,

if you solve it you get,

$\frac{\lambda(1-\lambda)}{2} \big( \langle Ax_1 , x_1 \rangle + \langle Ax_2 , x_2\rangle - \langle Ax_1 , x_2\rangle-\langle Ax_2 , x_1\rangle \big)$

now $\frac{\lambda(1-\lambda)}{2} > 0$ as $\lambda \in (0,1)$

and $ \langle Ax_1 , x_1\rangle + \langle Ax_2 , x_2\rangle - \langle Ax_1 , x_2\rangle-\langle Ax_2 , x_1\rangle \ = \ \langle A(x_1 - x _2), x_1 - x_2\rangle > 0$ as $A$ is positive definite