Talk:Singular value decomposition

message: Is there a difference between thin and truncated SVD? The description look as if it is the same. If there are differences, could someone mention them in the article? 129.187.173.178 (talk)  —Preceding undated comment added 17:31, 17 April 2010 (UTC)
thread_title:  Truncated vs. Thin SVD 

message: Just wanted to point out that the way SVD is presented in the article is the "Full" SVD formulation, which is different from the "Reduced" SVD formulation actually used in practice (ala Numerical Recipes).  The primary difference being that the reduced formulation has the U matrix chopped down to the same size as the original matrix, and the extra 0 rows in the singular value diagonal matrix likewise chopped off.  The Full form is favored in textbooks but not actually used anywhere in practice since the reduced form is easier to store, etc.  --65.198.5.139 (talk) 21:09, 21 August 2009 (UTC)
thread_title:  Full vs. Reduced SVD 

message: Reading this spectral derivation of the SVD I'm guessing that it would be fairly readable to somebody with first year linear algebra.  The exception is the injection of the use of the term isometry:
 "We see that this is almost the desired result, except that U1 and V1 are not unitary in general. U1 is a partial isometry (U1U*1 = = I ) while V1  is an isometry (V*1V1 = I ). To finish the argument, one simply has to "fill out" these matrices to obtain unitaries."
Reading the wikipedia page on partial isometry doesn't clarify things too much.  In that article there's references to:
Hilbert spaces, functional analysis, linear map, final and initial subspaces, isometric map.
It's all very abstract, much more so than ought to be required to read this section (I was looking for an explaination of SVD for nothing more than real matrices, so having to understand all of this stuff first would require a major digression).
I'm still figuring all this stuff out for myself so I'm not sure how one would rewrite this in a clear fashion (intuition says this would involve only references to projection and not isometries).  Perhaps somebody in the know already understands how this could be made clearer, and could do so.
Peeter.joot (talk) 15:42, 13 April 2008 (UTC)
replies: text: The article just means that U1 and V1 are not square matrices, so are not technically unitary.  However, they can be obviously or arbitrarily extended to unitary matrices by adding on some more orthonormal columns.  Basically you can ignore the sentence using the word isometry. JackSchmidt (talk) 15:52, 13 April 2008 (UTC)

text: Is the  = = in the statement, U_1^*U_1 = = I a typo?! Also, should it be U_1^*U_1 = I or U_1U_1^* = I ? I would have thought the former...  —Preceding unsigned comment added by 128.232.238.63 (talk) 21:10, 22 March 2009 (UTC)
thread_title:  References to isometries in SVD w/ spectral theorm section 

message: Young readers will be confused by the use of capital sigma for two different purposes, as the sign of summation, and as the diagonal matrix of eigenvalues.   When I learned this stuff 50 years ago they used lambda for the values. 171.66.187.197 (talk) 13:13, 1 July 2008 (UTC)
replies: text: I believe the notation used in the article is common practice. \Lambda can not be used, because singular values are not to be confused with eigenvalues. Also, \Sigma\, and \sum are easy to tell apart, especially since they are rarely used together. This only happens in section Separable models, because it uses capital sigma to denote singular values, which is inconsistent with the remaining article. I'm going to fix that now. --Drizzd (talk) 16:15, 2 July 2008 (UTC)
thread_title:  Confusing to use Capital Sigma as both variable and as summation 

message: Overall, I think the article on singular value decomposition is very clear and a great help in learning about this topic.  In the expression defining the pseudoinverse, it says:
M+=V \Sigma+U*, where \Sigma+ is the is the transpose of \Sigma with every nonzero entry replaced by its reciprocal.
However, the superscript '+' (plus) symbol attached to the M variable is not meant to modify M in the same way as \Sigma.  I would suggest a different superscript.
-- Richard C Yeh, 17 September 2008  —Preceding unsigned comment added by 204.124.117.40 (talk) 19:46, 17 September 2008 (UTC)
thread_title: Superscript '+' (plus) symbol meaning is overloaded in a confusing way

message: There is an 
  article in the New York Times Magazine of Nov 23, 2008.   It mentions that the SVD decomposition is the main tool for generating recommendations such as those in Netflix, but does not give details.   Perhaps somebody who understands this stuff could write a few paragraphs about this application? 
-- 194.24.158.8 (talk) 14:48, 28 December 2008 (UTC)
replies: text: That would be pretty cool. I started reading this article, because SVD was mentioned in one of the papers that the winners of the Netflix Prize published describing their solution. If I have time, I may go back and try to understand how SVD was used. The paper I was looking at also mentioned that other leading teams were using similar techniques, probably involving SVD. Danielx (talk) 02:46, 2 November 2009 (UTC)

text: I wouldn't be surprised if it also has to do with spectral clustering, which is used by Google's PageRank. —Ben FrantzDale (talk) 00:32, 4 May 2010 (UTC)

text: I believe that Netflix, in combination with other methods, uses latent factor models to perform collaborative filtering (i.e. predict user ratings and provide movie recommendations). Latent factor models are based on variations of the SVD in the same way as Latent semantic indexing. Latent semantic indexing, however, is latent factor methods applied to analysis of text. —Jon Starr  —Preceding undated comment added 15:07, 20 September 2011 (UTC)
thread_title:  Netflix 

message: Regarding the comment in "Low-rank matrix approximation": it seems more appropriate to be located on the talk page.  The original comment was:
"Note to editor: it would be likely that it should be diagonal, but it's not obvious to me that it must be diagonal... Why can't there be a exception where U* M V is not purely diagonal but still gives a minimal Frobenius norm?"
In response, it seems clear to me that, since Frobenius norm is the sum of the squares of each matrix element, the Frobenius norm in question could always be made smaller by zeroing any nonzero off-diagonal element of U^* \tilde M V since \Sigma is diagonal.  My question then is: can this possibly reduce the rank?
Phrank36 (talk) 19:09, 13 July 2010 (UTC)
thread_title:  Low-rank matrix approximation Proof 

message: This article is missing the singular value decomposition for operators acting between ∞-dimensional Hilbert spaces. (for example see R. Kress “Linear integral equations”.)  // stpasha »  08:36, 8 September 2010 (UTC)
thread_title:  SVD in ∞-dimensional spaces 

message: I nominated this article for A-class rating discussion (WP:WPM). You can discuss the new rating here.
I recently did a few adjustments to this article. Mainly, I edited the "Intuitive interpretations" section. However, my contribution was only a fraction of the work needed to make the article so complete and well written. The figure in the introduction, for instance, is an outstanding example of intuitively appealing geometrical interpretation, understandable by non-mathematicians. The introduction is simple, short, and effective. In my opinion, Eigenvalues and eigenvectors was worst than this article when it gained featured article status. Paolo.dL (talk) 15:11, 14 February 2011 (UTC)
thread_title:  This article deserves an higher rating 

message: The introduction animation is really useful.  However, I think the Sigma matrix should be written more explicitly. Currenly, it shows 4 elements, [S11, S12; S21, S22]
Since the Sigma matrix is automatically a diagonal matrix, perhaps the off-diagonal elements should be evaluated to zero, so that it is written as [S11, 0; 0, S22].
I think this will help further clarify what is already a great visualization.
thanks! 192.91.173.36 (talk) 12:42, 6 May 2011 (UTC)
thread_title:  Sigma matrix in graphical example 

message: While I recognise that this is a complex topic, I feel that the introductory section could be improved. As someone who is unfamiliar with mathematics, I would like to have a coarse-grained understanding of SVD. The article doesn't provide this for me. TimClicks (talk) 23:15, 25 June 2011 (UTC)
replies: text: How about inserting this sentence after the first sentence of the lead (partly based on a sentence from the figure caption): "If a matrix M is considered to be a transformation of n-dimensional space, the SVD makes M easier to describe by decomposing it into three simple transformations: a rotation V*, followed by a scaling (change of size) Σ along the rotated coordinate axes, then a second rotation U." ☺Coppertwig (talk) 15:57, 26 June 2011 (UTC)
replies: text:  The animated picture shows a similar sequence of simple transformations. I am not a mathematician, and I believe that the picture is extremely effective in providing intuitive insight about SVD. Your sentence, however, is not completely accurate. It is only true for square matrices with positive determinant. More information about this interpretation is given (purposedly) later, in the article (see Singular value decomposition#Intuitive interpretations). More generally, I believe that the current intro can hardly be improved, as it is already exhaustive and very well written. Also, I would not make it longer than it currently is.Paolo.dL (talk) 16:12, 27 June 2011 (UTC)
replies: text: OK. ☺Coppertwig (talk) 22:13, 10 July 2011 (UTC)
thread_title:  Introduction not very introductory 

message: I'm told there is a randomized algorithm for truncated SVD, that requires solving a matrix only slightly larger than the size of the desired output.  Could something be added to the article about this?  I do see there are some extlinks, that I'll look into.  Apparently the random sample has to be take rather carefully, but it makes it feasible to find truncated SVD's of very large matrices.  Thanks.
71.141.89.4 (talk) 08:55, 5 November 2011 (UTC)
replies: text: You may be thinking of the Lanczos algorithm (and block variants of it), which can be used to find the truncated SVD of very large matrices -- eg an SVD of the linearisation of an entire global weather-forecasting model's 24-hour evolution.  The algorithm is quite closely related to successively minimising a quadratic function by conjugate gradient.  Jheald (talk) 09:48, 5 November 2011 (UTC)
replies: text: Thanks, yes, that is probably what I heard about.   I see now that there is a paragraph about that in the article (mentioning weather prediction) though a little bit more exposition would be useful.  I'm interested in the subspace spanned by the singular vectors corresponding to (say) the largest 100 singular values of a very large matrix.  What I heard was that there was a way to carefully choose the random starting values so that by dealing with slightly more singular values than you actually want (e.g. you handle 110 dimensions instead of 100) you can get accurate output for your desired subspace.  The Lanczos algorithm article itself alludes to some methods for improving numerical stability, that might include what I'm describing, but it's not very explicit.  If I go read a book about numerical linear algebra, is it likely to cover stuff like this?  Thanks again.  71.141.89.4 (talk) 21:26, 6 November 2011 (UTC)

text: You are thinking of randomized SVD. It's a very hot topic right now, especially since the methods are well-suited to parallel and large scale problems. See the SIAM Review article by Halko, Martinsson and Tropp, http://arxiv.org/pdf/0909.4061.pdf 72.182.50.9 (talk) 21:39, 17 April 2016 (UTC)
thread_title:  randomized algorithm 

message: Why is the last matrix written as V* instead of V? Writing it this way seems to indicate that V itself has some importance, but it's never mentioned in the article.  — Preceding unsigned comment added by 69.205.120.39 (talk) 18:29, 6 February 2012 (UTC)
thread_title: V*

message: Following Wikipedia:WikiProject Mathematics/A-class rating/Singular value decomposition which was unsuccessful a few points may be helpful for editors wanting to improve the article.--Salix (talk): 09:11, 27 February 2012 (UTC)
thread_title: A-class rating

message: The caption beneath the animated figure at the upper right of the article states that it shows the SVD of a shearing matrix.  Part of it reads:
"The SVD decomposes M into three simple transformations: a rotation V*, a scaling Σ along the rotated coordinate axes and a second rotation U."
But the scaling depicted in the animation is not along the rotated coordinate axes, but rather along the original x- and y-axes.Daqu (talk) 19:56, 19 March 2012 (UTC)
thread_title:  Erroneous description of animated figure 

message: When doing a Google search for the term "singluar-value decomposition" (note hyphen), I cannot find a single instance where the hyphen is used in the first ten pages of search results. The term is not hyphenated. I will be moving the page back to "singular value decomposition" (note no hyphen) unless someone else does first. -- Schapel (talk) 11:35, 16 August 2012 (UTC)
thread_title:  The hyphen 

message: In the section "Calculating the SVD" it says "The SVD of a matrix M is typically computed by a two-step procedure. In the first step, the matrix is reduced to a bidiagonal matrix. "
I don't understand this reducing a matrix to a bidiagonal matrix.  What are we doing and why aren't we losing a lot of information?  RJFJR (talk) 18:14, 8 October 2012 (UTC)
replies: text: We are finding unitary X and unitary Y such that XMY*=B is bidiagonal. Then, more unitary matrices will be multiplied on the left and right to reduce B to diagonal form. Those, combined with the X and Y will be the final unitary matrices. Rschwieb (talk) 18:28, 8 October 2012 (UTC)
replies: text: I see.  Thank you.  Is there a way this can be added into the article?  RJFJR (talk) 19:34, 8 October 2012 (UTC)
thread_title:  Reducing matrix to bidiagonal 

message: The link to SVDLIBJ is broken. Ian S (talk) 11:03, 26 November 2012 (UTC)
thread_title:  Broken or obsolete link 

message: I propose the following change to the algorithm in iterative computation from its current form:
 \mathbf{r} = a random vector of length p
 do c times:
       \mathbf{s} = 0 (a vector of length p)
       for each row \mathbf{x} \in \mathbf{X}
             \mathbf{s} = \mathbf{s} + (\mathbf{x} \cdot \mathbf{r})\mathbf{x}
       \mathbf{r} = \frac{\mathbf{s}}{|\mathbf{s}|}
 return \mathbf{r}
to something like:
 \mathbf{r} = a random vector of length p
 do c times:
      \mathbf{r} = \mathbf{X}^{T} \cdot (\mathbf{X} \cdot \mathbf{r})
      \mathbf{r} = \frac{\mathbf{r}}{|\mathbf{r}|}
 return \mathbf{r}
And add something like: note the order of matrix multiplication.  By computing \mathbf{X}^{T} \cdot (\mathbf{X} \cdot \mathbf{r}), rather than (\mathbf{X}^{T} \cdot \mathbf{X}) \cdot \mathbf{r}, we avoid the Np^{2} computations involved in computing \mathbf{X}^{T}\mathbf{X}.
I'm a first time wikipedia author, so I wanted to ask for feedback rather than simply making a change.Haynorb (talk) 18:46, 24 June 2013 (UTC)
replies: text: I think this comment was intended to appear in the Principal Component Analysis talk page. (That pseudo-code does not currently appear on the Singular Value Decomposition page.)--Laughsinthestocks (talk) 21:12, 28 June 2013 (UTC)
thread_title:  Proposed Change to Iterative Computation 

message: The pseudoinverse of Σ is said to be "formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix".  Since Σ and hence the "resulting matrix" are diagonal it is redundant to transpose it. Ian S (talk) 11:48, 24 September 2013 (UTC)
thread_title:  Applications of the SVD/Pseudoinverse 

message: In the Example section, we see a SVD of the example matrix M, and we are shown the long form of each of U, Σ, and V*.
However, the subsequent paragraph purports to show us how U and V* are unitary, meaning (as I understand it) that the multiplication of each by its own _conjugate_ transpose yields an appropriate-size identity matrix.  As I followed through this, I expected to see "U" having the same form as "U" above.
Confusions:
1. In the working out of UU^T, the first matrix shown is not the same as U above (the second one worked out is U as shown above).
2. Shouldn't it be written as UU* though, since U is to be multiplied by its conjugate transpose, not merely its transpose (acknowledging that in the instant example, reals are used so the effect is moot)?
3. In the working out of VV^T, the opposite case is true: the first matrix shown corresponds to V* and not to V.  So looking at it tends to confuse same as (1.) above.
4. Echoing (2.) above, shouldn't it be written VV* (or, if (3.) is correctly understood, V*V)? 
I'm genuinely puzzled.  I know that due to the special case of using reals here and the special case of identity matrices, there is no difference in the putative result, but the whole point of the example is for an amateur like me to read along at home.
If I've got it wrong, let me know.  Otherwise, is it appropriate to reorder the symbols in the example for clarity? — Preceding unsigned comment added by 66.114.153.11 (talk • contribs) 
: You are correct about and 1 and 2. I'll work out the details to see if they are indeed unitary. For 4), since for an arbitrary unitary matrix , \mathbf{P} \mathbf{P}^{*} = I, it follows \mathbf{P}^{*} \mathbf{P} = I as  is square, rendering the order unimportant. TLA <FONT COLOR="#800000">3x ♭</FONT> → <FONT COLOR="#008000">♮</FONT> 19:06, 27 June 2014 (UTC)
replies: text: I think you are right about 3) too. Not that it matters much, but I think I will change that.Daniel (talk) 20:22, 23 September 2014 (UTC)
thread_title:  Confusing Example section? 

message: Say, it might be worthwhile to have a section that describes the connection between the SVD and the angle between subspaces.  If Q_1 is a matrix whose columns are orthonormal and span a subspace V_1\subset V, and Q_2 is a matrix whose columns are orthonormal and span a subspace V_2\subset V, then the largest singular value of Q_1^* Q_2 is the cosine of the angle between V_1 and V_2.
See https://en.wikipedia.org/wiki/Angles_between_flats#Angles_between_subspaces.  24.240.67.157 (talk) 00:34, 22 July 2015 (UTC)
thread_title:  Angle between subspaces 

message: The description of total least squares is incorrect; what it describes is actually Homogeneous least squares. Total least squares minimizes the Frobenius norm of [E | r] where b+r is in the range of A+E. See Golub and van Loan, Section 6.3 (4th ed.), or 12.3 (3rd ed.).  — Preceding unsigned comment added by 67.249.177.28 (talk) 22:35, 27 October 2015 (UTC)
thread_title:  Applications of the SVD/Total Least Squares 

message: The correct SVD of M is given below:
:\mathbf{M} = \begin{bmatrix}
                      1 & 0 & 0 & 0 & 2 \\
                      0 & 0 & 3 & 0 & 0 \\
                      0 & 0 & 0 & 0 & 0 \\
                      0 & 2 & 0 & 0 & 0
                    \end{bmatrix}
A singular value decomposition of this matrix is given by 
:\begin{align}
\mathbf{U} &= \begin{bmatrix}
               0 & 1 & 0 & 0 \\
 -1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 1 \\
 0 & 0 & -1 & 0 \\
             \end{bmatrix} \\
\boldsymbol{\Sigma} &= \begin{bmatrix}
                         3 & 0 & 0 & 0 & 0 \\
 0 & \sqrt{5} & 0 & 0 & 0 \\
 0 & 0 & 2 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 \\
                      \end{bmatrix} \\
\mathbf{V}^* &= \begin{bmatrix}
 0 & 0 & -1 & 0 & 0 \\
 \frac{1}{\sqrt{5}} & 0 & 0 & 0 & \frac{2}{\sqrt{5}} \\
 0 & -1 & 0 & 0 & 0 \\
 -\frac{2}{\sqrt{5}} & 0 & 0 & 0 & \frac{1}{\sqrt{5}} \\
 0 & 0 & 0 & 1 & 0 \\
               \end{bmatrix}
\end{align}
If one multiplies the matrices from the published example one gets 
\begin{bmatrix}
 0.894427 & 0. & 0. & 0. & 1.78885 \\
 0. & 0. & 2.23607 & 0. & 0. \\
 0. & 0. & 0. & 0. & 0. \\
 0. & 3. & 0. & 0. & 0. \\
               \end{bmatrix}
As of now the example agrees with you except for the order.  I agree with you--we should order by largest singular value, as described in the textChris2crawford (talk) 19:30, 17 March 2016 (UTC)
thread_title:  Example is wrong 

message: Currently the description of the SVD is completely nonsensical since it refers to non-square unitary matrices.
--75.65.100.154 (talk) 04:05, 10 February 2016 (UTC)
thread_title:  Incorrect Definition 

message: It's March 2016, and this page now includes a citation to an academic signal-processing reference from March 2016 saying that the SVD has been "successfully applied" to something or other. SVD is a basic tool used by many people, and I can't think of any useful reason for arbitrarily citing a brand-new paper. Much better to cite textbooks etc. Remove?--mcld (talk) 08:04, 29 February 2016 (UTC)
thread_title:  Possible self-citation issue 

message: The article states, "Thus the expression $UΣV∗$ can be intuitively interpreted as a composition of three geometrical transformations: a rotation or reflection, a scaling, and another rotation or reflection."  But, the previous sentence states,"$U, V∗$ can be viewed as rotation matrices."  So if $U$ and $V^*$ can be taken to be rotation matrices, then why does the first sentence I mentioned say "a rotation or reflection"?  I think this is an error.  — Preceding unsigned comment added by 2605:E000:8598:2A00:199C:889D:2DE2:3266 (talk) 10:16, 24 March 2017 (UTC)
thread_title:  The case where $M$ has positive determinant 

message: What is special with the V matrix that it has a star? i.e. why does SVD decompose into U sigma V* and not just U sigma V?
What does the star mean? Is there a way to turn V* into V, like if it's transposed or something?  — Preceding unsigned comment added by 90.209.48.196 (talk) 16:17, 17 July 2017 (UTC)
thread_title:  Why does the V matrix have a star? 

message: Is there a reason why we cannot just define it (I'm referring to the section Singular-value decomposition#Existence) as \Sigma=\begin{pmatrix}\mathbf{D}^{1/2}&\mathbf{0}_{r\times(m-r)}\\\mathbf{0}_{(n-r)\times r}&\mathbf{0}_{(n-r)\times(m-r)}\end{pmatrix}, where r is here the rank of M (and therefore of M^*M)? I think this would be less confusing than the way it is currently written. Luca (talk) 12:02, 25 April 2018 (UTC)
thread_title:  Definition of <math>\Sigma</math> confusing in construction section 

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

message: 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)
thread_title:  Look of Sigma-symbol representing the matrix holding the singular values 

message: 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)
thread_title:  Error in Figure 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Crucial error in first sentence? 

message: currently it says:
:\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
:\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)
thread_title:  Is Lagrange multiplier equation in existence proof Based on variational characterization correct?  

message: 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 Eckharhart 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)
thread_title:  History - reference to proof by Eckhart and Young