Let $\mathcal{P}$ denote the class of positive semidefinite symmetric bilinear forms $A : \mathbb{R}^{n} \times \mathbb{R}^n \to \mathbb{R}$. Let
$$
X(\mathcal{P}) := \{(x,y) \in \mathbb{R}^n \times \mathbb{R}^n \colon \, \forall A \in \mathcal{P}, \,\, A(x,y) \ge 0 \}.
$$ 
Clearly $\Delta := \{ (x, x)\in \mathbb{R}^n\times \mathbb{R}^n : x \in \mathbb{R}^n\}$ is a subset of $X(\mathcal{P})$. More generally, for scalars $\lambda, \mu \in \mathbb{R}$ having the same sign (that is, such that $\lambda \mu \ge 0$), we have $(\lambda \cdot x, \mu \cdot x) \in X(\mathcal{P})$. Here $\lambda \cdot x := (\lambda x\_1, \ldots, \lambda x\_n)$ for $x = (x\_1,\ldots, x\_n) \in \mathbb{R}^n$.


Does there exist a point in $X(\mathcal{P})$ not of the form $(\lambda \cdot x, \mu \cdot x)$ where $\lambda \mu \ge 0$? Is there an explicit description of $X(\mathcal{P})$? 



Nonnegative locus of the class of positive semidefinite symmetric bilinear forms
Maybe I am misunderstanding the problem because this solution looks simple. 


Essentially, you are asking if for every pair $x,y$ of linearly independent vectors there is a positive definite matrix $A$ such that $x^\*Ay<0$.


Let $x,y$ be two fixed linearly independent vectors. Take an invertible matrix $V$ whose first two columns are $x$ and $y$, so that $V^{-1}x=e\_1$, $V^{-1}y=e\_2$. Then, $x^\*Ay < 0$, where $A=V^{-\*}MV^{-1}$ and $M$ is any positive definite matrix with $M\_{1,2} < 0$.


