For a random matrix $A\in\mathbb{R}^{n\times m}$ with entries $A_{ij}\sim \mathcal{N}(0,1)$ i.i.d., the spectral norm of it can be bounded by $\|A\| \leq C(\sqrt{m} + \sqrt{n}+t)$ with probability $1-2e^{-t^2}$, where $C$ is some constant. This result is from Theorem 4.4.5 in High-Dimensional Probability by Roman Vershynin.
I am wondering why my analysis shows a different result. Here's my analysis.
By definition, the square of spectral norm of $A$ is:
$$\sup_{\|x\|=1,\|y\|=1} x^T A^TAy$$
Here each entry of $Ay$ and $Ax$ follows $\mathcal{N}(0,1)$, therefore, the above term follows $\chi^2(n)$. Or if we consider $AA^T$, it becomes $\chi^2(m)$. Then the spectral norm of $A$ is either $\sqrt{n}$ or $\sqrt{m}$. What goes wrong?
Thanks!