# Characterization of singular values

Let $$A \in \mathbb R^{m\times n}$$ and its singular values be denoted by $$\sigma_1 \geq \sigma_2 \geq \ldots \sigma_n \geq 0$$. Then $$\sigma_i(A) = \min\{\|B \|_2\colon B \in \mathbb R^{m \times n}, \mathrm{rank}(A-B) \leq i-1 \}.$$ How is this characterization of singular values obtained, is there a reference about it?

• This is a consequence of the Eckart-Young theorem. I'm not sure where to find a better reference for this fact, but I think "Topics in Matrix Analysis" by Horn and Johnson would be a good bet. – Omnomnomnom Feb 6 at 14:10
• I find the second proof given here particularly nice – Omnomnomnom Feb 6 at 14:19
• Does that answer your question sufficiently, or did you want a more thorough explanation? – Omnomnomnom Feb 6 at 19:00
• Clear, thanks for the reference! I got it through the first proof (optimality of the best rank-$i$ approximation by truncation of the SVD). – G. Gare Feb 7 at 12:17