I want to show that $\sigma_{k+1}$ is the $2$-norm (spectral) distance of $A$ to the nearest rank-$k$ matrix, with $\sigma_i$ denote the $i$-th singular value of $A$.

Could someone please give me some hints? Much appreciated. Thanks.

  • $\begingroup$ Thanks for your comments. I've edited my question. $\endgroup$ – Steve Dec 2 at 20:45

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