If we know that there are two vectors $x,y\in\mathbb{R}^d$ satisfying \begin{equation} \|x\|\ge c_1 \|y\|, \quad x^Ty\ge c_2\|x\|^2, \end{equation} where $c_1>0$ and $c_2>0$ are some given constants, can we always find a positive definite matrix $M\in\mathbb{R}^{d\times d}$ such that \begin{equation} x=My? \end{equation}