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Let $A\in\mathbb{R}^{n\times n}$. Let $s(A)$ be the number of entries of $A$ that are in $(-1,0)\cup(0, 1)$.

I am wondering is there any criteria (maybe some norms etc） implies that the size of $s(A)$ is small?

For example, if we consider the Frobenius norm $\sum_{i,j}A_{i,j}^2$, it cannot tell whether the matrix has small $s(A)$ but many zeros, or the matrix has large $s(A)$ but the entries are super close to zero. What if we assume the entries of $A$ are non-negative? Will it help?

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  • $\begingroup$ Do you have any assumptions on $A$? Invertible, normal, etc.? $\endgroup$ Commented Aug 15 at 4:55
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    $\begingroup$ @Daniel, I think Connor is asking whether there are any criteria which would imply there are few small entries. It sounds like everything – invertibility, normality, whatever – is on the table. Somewhat of a fishing expedition, I'd say. $\endgroup$ Commented Aug 15 at 10:02

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