Suppose we have a vector $e\in \mathbb{R}^n$, and a matrix $Q\in\mathbb{R}^{n\times p}$ such that $Q^{\top}Q=I_p$, i.e., the columns of $Q$ are orthornormal. Specifically, $e$ has i.i.d. subgaussian entries with mean zero and parameter $\sigma$.
Now I want to figure out the relationship between $\Vert Q^{\top}e\Vert_{2}$ and $\Vert e\Vert_{2}$ in order to find the Orlicz norm of the former norm. However, I get stuck in $$\Vert Q^{\top}e\Vert_{2}^2=e^{\top}QQ^{\top}e$$
and do not know what to do further. I want to derive a form like $$\Vert Q^{\top}e\Vert_{2}^2\leqslant c\Vert e\Vert_2^2.$$
And I want to ask does the constant $c$ depend on $n$? I appreciate for your help.
Since ${\bf Q} {\bf Q}^\top$ is a projection matrix, its eigenvalues are $0$ and $1$. Thus,
$$ \left\| {\bf Q}^\top {\bf v} \right\|_2^2 = {\bf v}^\top {\bf Q} {\bf Q}^\top {\bf v} \leq \underbrace{\lambda_{\max} \left( {\bf Q} {\bf Q}^\top \right)}_{= 1} \left\| {\bf v} \right\|_2^2 $$