I have two vectors $x$ and $y$ in $\mathbb{R}^n$. To rotate $x$ to $y$ we can use Householder transformation i.e. $Q=I-2 \frac{aa^T}{a^Ta}$ where $a=x-y$. Here $Q$ is the desired rotation matrix i.e. $Qx=y$. However, I am not sure about positive semi-definiteness of matrix $Q$. I also know, how to find closest p.d. matrix as shown in this journal (Higham's paper).
But now, I want to know a p.d. matrix $M$ such that $\|Mx-y\|$ is minimized. Does closest p.d matrix (as discussed in above link) to $Q$ is an answer to this problem ? If not, is there is a solution to above problem ?