Let $A\in\mathbb{R}^{k\times d}$ be matrix with i.i.d. $\mathcal{N}(0,1/k)$ entries with $k<d$, and let $B=A^{\top}A$. I would like to compute the distribution of $Bx$ where $x\in\mathbb{R}^{d}$ is a fixed vector.

The matrix $B$ essentially projects onto a $k$-dimensional subspace of $\mathbb{R}^{d}$ (it has rank $k$ almost surely). However, computing the distribution explicitly using the expression for entries of $Bx$ is tricky since the entries are dependent. Any hints on how to proceed?