$X^TXw$ is normal?

Question

I'm reading a paper in which it claims that if the matrix $X \in \mathbb{R}^{n \times n}$ has elements which are normal and independent, then for an arbitrary vector $w$, $X^TXw$ is distributed as $\sqrt{n}||w||N(0,I_n)$. Is it true? Because multiplying a normal to itself does not produce a normal. (http://papers.nips.cc/paper/6796-learning-relus-via-gradient-descent.pdf, page 7, equation 6.3)