Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2 \simeq n$ and the infinity norm squared is $\|A\|_{\infty}^2 = 1$. Is the following true?

$$\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2\gtrsim n$$

I cannot find a relation between matrix norms that can show this. For the spectral norm it is not true, as there is a nice construction.

Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2 \simeq n$ and the infinity norm squared is $\|A\|_{\infty}^2 = 1$. Is the following true?

$$\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2\gtrsim n$$

I cannot find a relation between matrix norms that can show this.

Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2\simeq n$, the infinity norm squared is $\|A\|_{\infty}^2=1$. Is it true that $\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2\gtrsim n$?

I cannot find a relation between matrix norms that can show this. For the spectral norm it is not true as there is a nice costrunction.

Thanks!

Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2 \simeq n$ and the infinity norm squared is $\|A\|_{\infty}^2 = 1$. Is the following true?

$$\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2\gtrsim n$$

I cannot find a relation between matrix norms that can show this. For the spectral norm it is not true, as there is a nice construction.

Let $A$ be a $n \times n$ matrix so that the Frobenius norm squared $\|A\|_F^2$ is $\Theta(n)$, the infinity norm squared $\|A\|_{\infty}^2=1$. Is it true that $\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2$ is $\Omega(n)$? Assume that $n$ is sufficiently large.

I cannot find a relation between matrix norms that can show this. For the spectral norm it is not true as there is a nice costrunction.

Thanks!

Let $A$ be a sequence of $n \times n$ matrices so that the Frobenius norm squared satisfies $\|A\|_F^2\simeq n$, the infinity norm squared is $\|A\|_{\infty}^2=1$. Is it true that $\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2\gtrsim n$?

I cannot find a relation between matrix norms that can show this. For the spectral norm it is not true as there is a nice costrunction.

Thanks!