  Johnson-Lindenstrauss guarantees certain topological structure is preserved
under random projections when project high dimensional deterministic vectors to
low dimensional vectors. In this work, we try to understand how random matrix
affect norms of random vectors. In particular we prove the distribution of the
norm of random vector $X \in \mathbb{R}^n$, whose entries are i.i.d. random
variables, is preserved by random projection $S:\mathbb{R}^n \to \mathbb{R}^m$.
More precisely, \[ \frac{X^TS^TSX - mn}{\sqrt{\sigma^2 m^2n+2mn^2}}
\xrightarrow[\quad m/n\to 0 \quad ]{ m,n\to \infty } \mathcal{N}(0,1) \] We
also prove a concentration of the random norm transformed by either random
projection or random embedding. Overall, our results showed random matrix has
low distortion for the norm of random vectors with i.i.d. entries.
