Talk:Orthogonal matrix

message: Surely that's incorrect. For an arbitrary noncanonical inner product <a, b> = a^T M b, a matrix Q preserves the inner product if and only if Q^T M Q = M. Unless M is (a multiple of) the identity, such matrices are not orthogonal. TotientDragooned (talk) 17:14, 2 August 2011 (UTC)
replies: text: The definition is with respect to a fixed inner product. It's not reasonable to expect a matrix to preserve all possible inner products. Mct mht (talk) 04:57, 3 August 2011 (UTC)
thread_title:  Orthogonal matrices preserve inner products?? 

message: Paul's Online Math Notes' (from Lamar University) pages on linear algebra have been taken off the Web.  They have been gone at least since January 2013 and are not there today (May 26, 2013).  The link to these notes should probably be removed from this article (and other articles if needed). Gsspradlin (talk) 02:46, 27 May 2013 (UTC)
thread_title:  Paul's Online Math Notes linear algebra section is down (between January and May 2013) 

message: Isn't the matrix 
:\begin{bmatrix}
2 & 0 \\
0 & \frac{1}{2}
\end{bmatrix}
a direct counterexample to the statement "Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1." that follows??? Is this statement correct?
(Bilingsley (talk) 14:29, 21 January 2016 (UTC)
replies: text: Your matrix isnt orthogonal, so the statement does not apply.  — Preceding unsigned comment added by 2601:19B:B00:87A0:F574:DC63:8623:4EB0 (talk) 00:19, 7 March 2021 (UTC)
replies: text:  That matrix Bilingsley presents is an orthogonal matrix. Its vectors are orthogonal. Aaronfranke (talk) 17:45, 13 October 2023 (UTC)
thread_title:  Properties 

message: The first few lines state:
"An orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors),"
However I've always been a bit confused about this. Surely the definition in terms of matrices is only true when the basis is itself orthonormal? In components
(QTQ)ij = (QT)ikQkj = QkiQkj = (Qi)k(Qj)k where repeated indices are summed. But the inner product of Qi and Qj is equal to (Qi,Qj) = (Qi)n(Qi)m(en,em). Only when the basis is orthonormal and (en,em) = δij can the usual 'dot product' inner product be used. At least that is how it seems to me.  — Preceding unsigned comment added by Oward98 (talk • contribs) 10:54, 19 August 2018 (UTC)
thread_title: Introduction

message: If these two terms mean the same thing, should we add the other terminology in the lead? E.g. Orthogonal matrix, or orthonormal matrix, etc.  — Preceding unsigned comment added by MurrayScience (talk • contribs) 13:44, 24 January 2021 (UTC)
thread_title:  Orthogonal matrix =? Orthonormal matrix 

message: This article makes two factually inaccurate statements in the opening sentences: (i) that orthogonal matrices must be real, and (ii) that the extension of the property of orthogonality to complex matrices is unitarity. Both are incorrect: orthogonality is the requirement that a matrix satisfy O^T O = I, and makes no requirement regarding the field from which the elements of O are drawn. Moreover, the aforementioned definition of orthogonality can be applied to complex matrices as is, and is not equivalent to unitarity.
While orthogonal matrices with non-real elements are less commonly occurring in mathematics and the sciences than real orthogonal matrices, they do occur, and there is a literature which studies their properties (eg Choudhury & Horn, and Horn & Merino). An extensive clean up seems to be required to make apparent how much of the content of this article actually applies in general to orthogonal matrices, and how much is results special to the real-orthogonal case.  — Preceding unsigned comment added by 2601:19B:B00:87A0:7108:9F93:3ED6:8DB0 (talk) 15:09, 3 March 2021 (UTC)
thread_title:  Orthogonal matrices need not be real 

message: Hello, this is my first ever comment on a Wikipedia mathematics article.
The LaTeX formatting is not working correctly throughout this article. For example see the superscipted T indices near the top of the article: 
Instead of displaying as Q{^T}Q = QQ^{T}
It displays as $Q^{\mathrm {T}}Q = QQ^{\mathrm {T}}=I,$
Note this only occurs on this math article, I read many other articles with LaTeX embedded with no problems. So I doubt it's a browser issue (I use FireFox).
I do have a JPG screenshot/snippet showing the issue, but Wikipedia would not let me upload it. 
Curiously the problem is not visible when I open the "Edit" tab where the formatting works just fine, so I'm not sure how to offer an edit / fix.
CJ7903 (talk) 22:25, 26 December 2021 (UTC)
thread_title:  Latex formatting error 

message: Orthogonal means that the vectors are perpendicular. See Orthogonality and Orthogonality_(mathematics). Orthonormal is a special case of orthogonal where the vectors are both orthogonal ("ortho") and normalized ("normal"). The current article is incorrect to equate these terms. Aaronfranke (talk) 17:35, 13 October 2023 (UTC)
thread_title:  Orthogonal and orthonormal are not the same thing 

message: :The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.
The result of the move request was: Not moved, consensus is that the current title is the COMMONNAME of the topic (non-admin closure) BegbertBiggs (talk) 13:58, 21 October 2023 (UTC)
replies: text: Orthogonal matrix →  – This page describes orthonormal matrices, which a special case of orthogonal matrices, but are not the same. Orthonormal means orthogonal ("ortho") and normalized ("normal"). Orthogonal is a well-defined term, orthogonal does not mean normalized, orthogonal only means orthogonal. After the move, orthogonal can get its own page. Aaronfranke (talk) 17:54, 13 October 2023 (UTC)

text: Oppose per WP:COMMONNAME. We don't use Wikipedia to prescribe or recommend the "correct" terminology, but reflect the common usage, see e.g. https://netspeak.org/#q=%5Borthogonal+orthonormal%5D+matrix or compare :google:"orthogonal matrix" and :google:"orthonormal matrix". I totally agree that orthonormal matrix could have been a better name, for reasons Aaronfranke mentioned, but this is not given any weight by WP:NC until its prevalence in reliable sources.—HTinC23 (talk) 17:26, 14 October 2023 (UTC)
replies: text:  I disagree. It's not about prescribing, it's about following definitions: "adjective word" should never mean anything other than "word that is adjective" (there are exceptions like dictatorships calling themselves "Democratic People's Republic", but in those cases some entity decides its own name, nobody owns math). My argument is that the existing sources do not matter, they are wrong. In my opinion it is irrelevant that Charles Hermite used this term in 1854. To me it's like arguing WP:SKYBLUE, words can't simply be redefined. Aaronfranke (talk) 19:03, 14 October 2023 (UTC)

text: Oppose per the Adumbrativus and HTinC23.  This is not a close question, please find something better to do with your time.  --JBL (talk) 19:36, 14 October 2023 (UTC)

text: Oppose, (again) because the current title is the common name for this concept. Yes, I agree the inconsistency is annoying, but it's what we have at the moment. To be frank, claiming that  shows a fundamental misunderstanding of what Wikipedia is supposed to do. Wikipedia summarizes information that appears in reliable sources; including information is a matter of verifiability, not truth. Anon126 (notify me of responses! / talk / contribs) 11:42, 21 October 2023 (UTC)
thread_title:  Requested move 13 October 2023