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I am interested in the following problem: Let $A$ be a $n \times n$ matrix with columns that have norm $\sqrt{n}$, can we say something about the norm of its rows? In particular can we bound these norms by a function of $n$?

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  • $\begingroup$ All entries are bounded by the norm of their column; how could the row norms not be bounded? $\endgroup$ Commented Dec 5, 2021 at 17:34
  • $\begingroup$ The word "norm" could many many things. Your title uses "norm of ... columns" twice, while the body asks about the norm of rows. Please edit your Question and explain the specific context in which $n$ represents both the shape of the matrix and $\sqrt n$ the norm of columns. $\endgroup$
    – hardmath
    Commented Dec 8, 2021 at 0:57

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Let $X_1, \dotsc, X_n$ be the columns and consider maximising norm of the vector $A=(X_{11}, \dotsc, X_{n1})^T$. $\|A\|^2 = \sum_{j=1}^n(X_{j1}^2)$, so we want to maximise each $X_{j1}$ subject to the constraint $\|X_j\|=\sqrt{n}$. The maximum is when each $X_{j1}=\sqrt{n}$ with $\|A\| = n$.

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