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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_n \quad [\mathbf{A}\mathbf{A}^\mathrm{H}]_{n,n},\\ &\max_n \quad [\bar{\mathbf{A}}\bar{\mathbf{A}}^\mathrm{H}]_{n,n}, \end{align}\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]_{n,n}$$[\cdot]_{m,m}$ is the $n$$m$th diagonal entry of $\mathbf{A}$ and $(\cdot)^{\mathrm{H}}$ denotes conjugate transpose operation.

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_n \quad [\mathbf{A}\mathbf{A}^\mathrm{H}]_{n,n},\\ &\max_n \quad [\bar{\mathbf{A}}\bar{\mathbf{A}}^\mathrm{H}]_{n,n}, \end{align} where $[\cdot]_{n,n}$ is the $n$th diagonal entry of $\mathbf{A}$ and $(\cdot)^{\mathrm{H}}$ denotes conjugate transpose operation.

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.

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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_n \quad [\mathbf{A}\mathbf{A}^\mathrm{H}]_{n,n},\\ &\max_n \quad [\bar{\mathbf{A}}\bar{\mathbf{A}}^\mathrm{H}]_{n,n}, \end{align} where $[\cdot]_{n,n}$ is the $n$th diagonal entry of $\mathbf{A}$ and $(\cdot)^{\mathrm{H}}$ denotes conjugate transpose operation.