Talk:Singular value decomposition

In the entire article, it is not clear to me whether ${\displaystyle V^{*}}$ means the conjugate transpose of ${\displaystyle V}$, or just the complex conjugate. I guess it doesn't really matter, because if ${\displaystyle V}$ is unitary, so are the conjugate of ${\displaystyle V}$ and the conjugate transpose of ${\displaystyle V}$. But in any case, I'm confused. In the definition, should we add something like " ..., and ${\displaystyle V^{*}}$ is the complex conjugate of ${\displaystyle V}$ " ? — Preceding unsigned comment added by Barbireau (talk • contribs) 13:16, 1 April 2020 (UTC)

Hello I'm another user adding that agrees, but is unsure for what V* is, also maybe a voting measure or addable box for comments in discussions would be beneficial! — Preceding unsigned comment added by DaltSalt (talk • contribs) 11:20, 12 May 2020 (UTC)

I have just replaced every (capital) Sigma representing the matrix holding the singular values that I could find in the article (apart from captions) with the version of the first appearance in the text that also matches the height of the unitary matrix symbols frequently used next to it. (And what many ways there are to code it, and miscode it for that matter; it took me over half an hour to figure out how to code it geared to the surroundings where it appears.) But at least, if I succeeded, there is now unity of notational appearance in this respect.Redav (talk) 13:26, 23 May 2020 (UTC)

There is an error in the figure. The sigma's should align with the U vectors. Here is an example of a correct figure http://i.stack.imgur.com/IM6Fn.png — Preceding unsigned comment added by 73.12.205.109 (talk) 19:40, 4 February 2021 (UTC)

The opening sentence seems to claim that only normal matrices are diagonalizable. That claim is false, right? What's the fix? Or is there a reliable source that can explain it to me? Mgnbar (talk) 00:39, 27 July 2021 (UTC)

The claim is talking about the special eigendecomposition of a normal matrix where one can say that the matrix is unitarily similar to a diagonal matrix, not merely similar. Normal matrices are (definitionally) the only ones that can be decomposed in this way. I tried to make the lead more correct, but the wording is a bit clunky. Fawly (talk) 01:19, 27 July 2021 (UTC)

Thank you. I understand the intent much better now. I have further broken the big, complicated sentence into small sentences, hopefully for the better. Mgnbar (talk) 12:03, 27 July 2021 (UTC)

currently it says:

${\displaystyle \nabla \sigma =\nabla \mathbf {u} ^{\textsf {T}}\mathbf {M} \mathbf {v} -\lambda _{1}\cdot \nabla \mathbf {u} ^{\textsf {T}}\mathbf {u} -\lambda _{2}\cdot \nabla \mathbf {v} ^{\textsf {T}}\mathbf {v} }$.

I don't quite know what this means, I would think it should be

${\displaystyle \nabla \mathbf {u} ^{\textsf {T}}\mathbf {M} \mathbf {v} -\lambda _{1}\cdot \nabla \mathbf {u} ^{\textsf {T}}\mathbf {u} -\lambda _{2}\cdot \nabla \mathbf {v} ^{\textsf {T}}\mathbf {v} =0}$

instead to match the eigenvalue version as well as the following equations. 2001:56A:F98E:2400:6C3F:8EA:5F2:540A (talk) 21:23, 6 November 2022 (UTC)

The article claims

The first proof of the singular value decomposition for rectangular and complex matrices seems to be by Carl Eckart and Gale J. Young in 1936;

but their paper states up front:

The solution of the problem is much simplified by an appeal to two theorems which are generalizations of well known theorems on square matrices.[here is given a citation to (Courant and Hilbert 1924) getting SVD for square matrices] They will not be proven here.

Theorem I. For any real matrix a, two orthogonal matrices u and U can be found so that lambda = uaU' is a real diagonal matrix with no negative elements.

This is their statement of the purely algebraic fact about SVD (the problem they are considering is about approximating matrices, which is not what the main article is about), and not proved! The usual statement is given immediately following the theorem in equation (10). It might be that Eckart and Young first stated SVD for non-square matrices, but false to say they gave a proof. It's also not true that they considered the complex case; the application they have in mind is purely real, and nowhere do they mention complex entries. 121.44.213.90 (talk) 07:43, 19 July 2023 (UTC)

Reference 4 link is dead to me. Try https://people.wou.edu/~beavers/Talks/Willamette1106.pdf SAM26 (talk) 20:34, 20 September 2025 (UTC)

I put this link in, although it's not a great source. Tito Omburo (talk) 20:40, 20 September 2025 (UTC)