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In the 4th edition of Linear Algebra Done Right by Sheldon Axler, the Singular Value Decomposition is stated and proved on page 287. With the way the proof is constructed, shouldn't it have been split for $\underline {T \ \text{is injective} \ }$ or $\underline {T \ \text{is not injective} \ }$? I'm asking this because the proof at some point mentions that

If $k \in \{1, \ldots, n\}$ and $k>m$, then $s_k = 0$

but this means that $0$ is a singular value of $T$, implying that $T$ is not injective. But we want to prove the Singular Value Decomposition for any linear map $T$.

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  • $\begingroup$ @AnneBauval added the "not", thanks! How should I make it self-contained? I linked to the exact claim and proof. $\endgroup$ Commented Apr 13 at 4:18
  • $\begingroup$ I meant: avoid links. Uploading 2 Mo on my phone bothered me. However, I did. $\endgroup$ Commented Apr 13 at 4:25

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There is no need to treat the injective case separately. If $m=n$, there is no $k \in \{1, \ldots, n\}$ such that $k>m$, but this does not invalidate the proof.

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    $\begingroup$ Ah, you cleared it up for me. Thank you! $\endgroup$ Commented Apr 13 at 4:26

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