# Effect of column normalization on maximum diagonal entry

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.

Let $$a_{m,n}$$ be the elements of matrix $$\mathbf{A}$$. Then, we have \begin{align} \beta_{\max}&=\max_m \quad [\mathbf{A}\mathbf{A}^\mathrm{H}]_{m,m}=\max_m \sum_{n=1}^{N}|a_{m,n}|^2\\ \bar{\beta}_{\max}&=\max_m \quad [\bar{\mathbf{A}}\bar{\mathbf{A}}^\mathrm{H}]_{m,m}=\max_m \sum_{n=1}^{N}\frac{|a_{m,n}|^2}{\|\mathbf{a}_n\|^2}. \end{align} Therefore, by defining $$\alpha_{\max}=\max_n \|\mathbf{a}_n\|$$ and $$\alpha_{\min}=\min_n \|\mathbf{a}_n\|$$, we have \begin{align} \bar{\beta}_{\max}&=\max_m \sum_{n=1}^{N}\frac{|a_{m,n}|^2}{\|\mathbf{a}_n\|^2}\\ &\leq \max_m \sum_{n=1}^{N}\frac{|a_{m,n}|^2}{\alpha_{\min}^2}\\ &=\frac{\beta_{\max}}{\alpha_{\min}^2}. \end{align} Similarly, we obtain \begin{align} \bar{\beta}_{\max}\geq\frac{\beta_{\max}}{\alpha_{\max}^2}. \end{align}