Talk:Orthogonal matrix

Orthonormal matrix

Section moved from Talk:orthonormal matrix -- Jitse Niesen (talk) 14:55, 2 June 2006 (UTC)

I do not think the term orthonormal matrix is standard anywhere. Perhaps someone created this page in an effort to popularize the term. Michael Hardy 03:18, 22 March 2003 (UTC)

It is indeed nonstandard, so I merged the article orthogonal matrix with this article. However, I do seem to remember having seen it used somewhere (hmm, that's a contorted verbal construction). -- Jitse Niesen (talk) 14:55, 2 June 2006 (UTC)

I disagree. In my current courses, Continuum Mechanics and Fluid Mechanics at the University of Minnesota the term orthonormal is used quite often and found in our textbook and in a few research papers. In particular "Introduction to the Mechanics of a Continuous Medium" by Lawrence E. Malvern. It is also noted on http://mathworld.wolfram.com/OrthonormalBasis.html I've even had one assignment requesting I find orthonormal eigen vectors. The manner in which it is mentioned in my courses implies it is standard in the field of Aerospace Engineering. -- Kruzicka (talk) 16:55, 28 September 2006 (UTC)

The term "orthonormal" is completely standard in the phrase "orthonormal basis". And, confusingly, the columns of an "orthogonal matrix" do comprise an orthonormal basis. Despite this overlap, the phrase "orthonormal matrix" is at odds with standard mathematical practice, including applied mathematics and physics. To quote from the article,

• A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space Rⁿ with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of Rⁿ. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy M^TM = D, with D a diagonal matrix.

Since you misinterpret the content of the MathWorld site, which I can see, I'm inclined to suspect you make the same mistake with the Malvern text, which I cannot see. --KSmrq^T 22:53, 28 September 2006 (UTC)

I've seen "othronormal" used in courses as well. It may be a less popular alternative. I'll make a note as such. Danielx (talk) 00:34, 2 November 2009 (UTC)

i would assume the line "An orthogonal matrix is a special orthogonal matrix if its determinant is +1" at the start is ment to be "An orthonormal matrix is a special orthogonal matrix if its determinant is +1" as having the sentance that "A is a special case of A" isnt really saying anything, so im changing it Shinigami Josh (talk) 11:40, 22 October 2008 (UTC)

The sentence "An orthogonal matrix is a special orthogonal matrix if its determinant is +1" makes perfectly sense to me, but perhaps it's easy to misread it, so I reformulated it. -- Jitse Niesen (talk) 12:01, 22 October 2008 (UTC)

Since $det(A^{T})=det(A)$, $det(AB)=det(A)det(B)$, and $QQ^{T}=Id$ if Q is orthogonal, we can see that $det(Id)=1=det(QQ^{T})=det(Q)det(Q^{T})=det(Q)^{2}=1$. So Either $det(Q)=1$ or = $-1$. —Preceding unsigned comment added by 84.78.203.218 (talk) 17:24, 7 March 2010 (UTC)

Isn't the introduction somewhat ambiguous? It says "In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors)." If $Q$ is an orthogonal matrix then according to the current version $cQ$ must also be an orthogonal matrix because the vectors are still orthogonal(but no longer orthonormal) Ranadeakshay (talk) 20:59, 1 April 2014 (UTC)

Orthogonal matrices preserve inner products??

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)

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)

Paul's Online Math Notes linear algebra section is down (between January and May 2013)

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)

Properties

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))

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)

That matrix Bilingsley presents is an orthogonal matrix. Its vectors are orthogonal. Aaronfranke (talk) 17:45, 13 October 2023 (UTC)

Introduction

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

(Q^TQ)_ij = (Q^T)_ikQ_kj = Q_kiQ_kj = (Q_i)_k(Q_j)_k where repeated indices are summed. But the inner product of Q_i and Q_j is equal to (Q_i,Q_j) = (Q_i)_n(Q_i)_m(e_n,e_m). Only when the basis is orthonormal and (e_n,e_m) = δ_ij 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)

Orthogonal matrix =? Orthonormal matrix

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)

Orthogonal matrices need not be real

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)

Latex formatting error

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)

Orthogonal and orthonormal are not the same thing

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)

Requested move 13 October 2023

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)

---

Orthogonal matrix → Orthonormal 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)

• Oppose. The term "orthogonal matrix" is overwhelmingly standard; WP:COMMONNAME applies. See Strang, Introduction to Linear Algebra section 4.4, which flat-out says: 'The inverse is the transpose. In this square case we call Q an orthogonal matrix. [Footnote:] "Orthonormal matrix" would have been a better name for Q, but it's not used.' To say it's never used would be hyperbole, but the point remains that it's not the common name. It may be frustrating that the same word ("orthogonal") in slightly different contexts can mean different things, but that's just the way it is. Adumbrativus (talk) 01:07, 14 October 2023 (UTC)

  You can't just redefine "orthogonal", no matter how common it is to misuse this term. It does not make sense for "orthogonal matrix" to mean anything other than "a matrix that is orthogonal". This is similar to redefining "kilobyte" to be 1024 bytes instead of 1000 bytes. You can't just do that. This is why we created new terminology: Ki, so KiB for 1024 bytes. Similarly, the only correct terminology to refer to a matrix that is orthonormal is "orthonormal matrix". Aaronfranke (talk) 02:58, 14 October 2023 (UTC)

  You can't just redefine "orthogonal", no matter how common it is to misuse this term. This is a weird idea: of course you can define the word "orthogonal" to have two different meanings simultaneously; it is an extremely common thing to do with a word, both inside and outside of mathematics. --JBL (talk) 19:34, 14 October 2023 (UTC)

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)

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)

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)

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 the existing sources do not matter, they are wrong 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)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.