I have been reading Vershynin's "High-Dimensional Probability," and have a question regarding Theorem 4.4.5, which I will include first:
Theorem 4.4.5 (Norm of matrices with sub-gaussian entries)
Let A be an $m \times n$ random matrix whose entries $A_{ij}$ are independent mean-zero sub-gaussian random variables. Then, for any $t > 0$ we have $$ \|A\| \leq CK(\sqrt{m} + \sqrt{n} + t) $$ with probability at least $1-2\exp(-t^2)$. Here $K = \max_{i,j}\|A_{ij}\|_{\psi_2}$ and $C$ is a positive absolute constant.
Now, my question is this:
Can I reformulate this theorem to only have $t$ on the right side of the inequality? In other words, is it possible to use some clever substitution to get $C, K, m, n$ to all be in the argument of the exponential function instead?
(This can for example be done for Bernstein's inequality, see this question for example: Manipulating Concentration Inequalities, but I can't figure out the substitution here)