Let $Q$ be a real symmetric positive semidefinite $n \times n$ matrix. Consider a set

$$
\Big\{ x \in \mathbb{R}^n \;\Big| \; x^{\rm T}Qx\le 1\Big\},
$$

which can be loosely described as an "elliptic cylinder". (It would be an ellipsoid if $Q$ was positive definite).

**Question.** What is the image of this set under a linear map $y = Cx$? One can assume that $C$ has full row rank, but no more than that.

I think that it will be
$$
\Big\{ y \in \mathbb{R}^m \;\Big| \; y^{\rm T}Ry\le 1\Big\},
$$

where $R$ is some positive semidefinite matrix. But that is not enough: I actually want to find an *explicit formula* for $R$ (in terms of $Q$ and $C$) $-$ as elementary as it can be.