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If we randomly pick $k\ll n$ columns from a fixed $n\times n$ matrix $A$, what can one say about the distribution of the 2-norm condition number of the resulting $n\times k$ matrices $A_k$?

I'd expect that knowing the singular values of $A$ would strongly restrict the distribution. (If all singular values of $A$ are identical, then this also holds for the $A_k$.)

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  • $\begingroup$ have you considered the case of a diagonal matrix? in this case, the singular values are just which entries end up being picked. $\endgroup$ Commented Apr 27, 2021 at 14:46
  • $\begingroup$ Unfortunately this tells very little about the general case. $\endgroup$ Commented Apr 27, 2021 at 14:57

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