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}



Find a matrix transformation for 2 vectors with conditions
Yes. We only need $x^T y > 0$, which is implied by your conditions if neither x nor y is zero. Let $s$ be the positive square root of $x^T y$, and $A$ a $d\times d$ real matrix who's first column equals $\frac1s x$. Let the remaining columns be a basis for the codimension-$1$ space perpendicular to y. Let $e\_1$ be the first standard basis vector.


As x is not perpendicular to y, $A$ is non-singular, and $A \left(s e\_1\right) = x$.


Also $A^T y = \left(\frac1s x^T y, 0, \dots, 0\right)^T = s e\_1$.


Therefore $A A^T y = A \left(s e\_1 \right) = x$. We let $M = A A^T$, which is positive definite.


