Let $\mathbf{A}$ be a $M\times N$ complex matrix, and $\bar{\mathbf{A}}$ be constituted by normalizing each column of $\mathbf{A}$. Therefore, we have
$$\mathbf{A}=\bar{\mathbf{A}}\mathbf{\Gamma},$$
where $\mathbf{\Gamma}=\mathrm{diag}(\|\mathbf{a}_n\|)$ is a diagonal matrix and $\mathbf{a}_n$ is the $n$th column of matrix $\mathbf{A}$. Is there any relation between these two following quantities:
\begin{align}
&\max_m \quad [\mathbf{A}\mathbf{A}^\mathrm{H}]_{m,m},\\
&\max_m \quad [\bar{\mathbf{A}}\bar{\mathbf{A}}^\mathrm{H}]_{m,m},
\end{align}
where $[\cdot]_{m,m}$ is the $m$th diagonal entry of $\mathbf{A}$ and $(\cdot)^{\mathrm{H}}$ denotes conjugate transpose operation.