I'm dealing with two positive definite matrices $A,B\in\mathcal{M}(\mathbb{R}^{d\times d})$, such that $A \geq B$ pointwise. I want to know for which vectors $u,v\in\Bbb R^d$ we have the following inequality:
$$ u^T B v \leq u^T A v. $$
Now obviously, if $u,v\geq 0$ pointwise, we have
$$ u^T B v = \sum_{i,j=1}^d u_i B_{ij} v_j \leq \sum_{i,j=1}^d u_i A_{ij} v_j = u^T A v. $$
I'm wondering now if there are more general conditions on $u,v$ for the above to hold. Can someone point me in the right direction? Thanks in advance.
For simple example, we let $B=I$ and $A ={\rm diag}\ (R,1),\ R>1$.
If $u=(1,0),\ v=e^{it},\ -\frac{\pi}{2}\leq t\leq \frac{\pi}{2}$, then $u\cdot v\leq u^T Av$.