Talk:Orthogonality

message: Those trained in computer science think they invented everything known before computers existed: integrals, mathematical induction, orthogonality, etc. I've left the page a bit of a messy hodge-podge, but far better than what was here.  Michael Hardy 02:27, 13 Jan 2004 (UTC)
replies: text: If non-orthodox is "heterodox", is "heterogonal" non-orthogonal? (Google has one hit for that word, in an unmaths context.) 142.177.126.230 21:05, 5 Aug 2004 (UTC)
replies: text: It is a needless complication of the definition of orthogonality to bring in the subscripts i and j when one is only trying to define what it means to say that two functions are orthogonal. Also, it is incorrect, unless one has first given the subscripts some meaning. Michael Hardy 01:45, 6 Sep 2004 (UTC)
thread_title:  Orthogonality existed long before computers! 

message: There is a missing opening square bracket on the integration example image, I believe. --anon
: Fixed now. I think that bracket was left out on purpose. But I agree with you that things look better with the bracket in. Oleg Alexandrov 18:25, 15 May 2005 (UTC)
thread_title: Missing bracket

message: This article does not mention orthogonal curves or explain what it means that two circles are orthogonal to each other. Hyperbolic geometry mentions orthogonal circles, but I had to look up the exact meaning elsewhere (more precisely, on MathWorld).
My question is, should orthogonal curves and circles be covered in this article, or do they qualify as a "related topic"? Fredrik | talk 03:16, 21 Oct 2004 (UTC)
replies: text: The concept is not really that different, though Mathworld's geometric treatment may merit a seperate page. One could perhaps say generally that two curves parametrized by functions f and g are orthogonal, if where they interesect ∇f.∇g = 0, though I'm not sure that's a decent, established, or useful definition... Dysprosia 08:14, 21 Oct 2004 (UTC)
replies: text: A section on orthogonal curves must certainly be added to the article.--Shahab (talk) 08:43, 8 March 2008 (UTC)
thread_title: Orthogonal curves

message: I have just put in an emergency fix for the question raised by 66.91.134.99, and left a note on his talk page. This was a proof that an orthogonal set is a linearly independent set.
It's not at all clear that putting in this proof is the right thing for the article as a whole -- I just needed a quick fix. (It's not even clear that this is the
best proof. It was off the top of my head. And it is definitely not formatted well.)  Maybe the linear independence is truly obvious, and saying anything about it is just inappropriate for the level of the discussion. Maybe the proof/discussion should be elsewhere.
If/when someone has the time to look over the whole article, and think about the context of the orthogonality/independence issue, and figure out the right way to deal with all this, it would be a big help.
William Ackerman 16:08, 21 July 2006 (UTC)
replies: text: Thanks for the proof. But I tend to agree with your doubt that the proof was not the right thing for the article as a whole, especially that early in the article. Proofs are not really encyclopedic to start with (see also Wikipedia:WikiProject Mathematics/Proofs). I removed the proof for now.Oleg Alexandrov (talk) 08:33, 22 July 2006 (UTC)
thread_title:  Emergency fix 

message: We say that two variables are orthogonal if they are independent. Uncorrelated seems much more plausible, since the distribution of the product of two variables is an inner product of the variables.Septentrionalis PMAnderson 05:27, 19 September 2007 (UTC)
replies: text: Should we give more prominence to the section on statistics in this article? My discipline is psychology, and if I were to refer to two variables as orthogonal, I would mean that they are not statistically significantly correlated. I would probably add a wikilink to orthogonal, so that curious readers could find out what this term means by going to this article, rather than defining the term in every article for which I use this word. However, at the moment, this article is rather heavy for a non-specialist. ACEOREVIVED 19:23, 13 October 2007 (UTC)
replies: text: That's right: two random variables are called orthogonal if they're simply uncorrelated, not necessarily independent. I made the appropriate change. There's often confusion about this because for Gaussian random variables, uncorrelatedness implies independence. For general r.v.'s, though, this isn't true: independence is a stronger condition. Jgmakin 06:37, 25 October 2007 (UTC)
thread_title: Statistics

message: Here is the current intro:
In mathematics, orthogonal, as a simple adjective not part of a longer phrase, is a generalization of perpendicular. It means "at right angles". The word comes from the Greek  (orthos), meaning "straight", and  (gonia), meaning "angle". Two streets that cross each other at a right angle are orthogonal to one another. In recent years, "perpendicular" has come to be used more in relation to right angles outside of a coordinate plane context, whereas "orthogonal" is used when discussing vectors or coordinate geometry.
I propose this replacement:
In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the  Greek  (orthos), meaning "straight", and  (gonia), meaning "angle". For example, a subway and the street above, although they do not physically intersect, are orthogonal if they meet at a right angle.
This version avoids the need to say that "orthogonal" is a generalization of "perpendicular" by saying that they are identical in the generalized context of vector mathematics. The example is improved by not using the coordinate plane. Lastly, the note about common usage is redundant because the intro sentence has already described the context of "orthogonal". --Beefyt (talk) 05:52, 4 August 2008 (UTC)
replies: text: From beginning Euclidean geometry, two lines (straight) that do not intersect are "skew" [False. In plane geometry, two lines either intersect or they are parallel to one another. There are no any other possibilities.] "Perpendicular" implies the lines intersect in a right angle. "Orthogonal" implies far more than mere perpendicularity, as evidenced in the article and its attendant commentaries.  —Preceding unsigned comment added by Lionum (talk • contribs) 06:46, 11 October 2008 (UTC)
replies: text: But the lead says "two vectors are orthogonal if they are perpendicular", not "two lines are orthogonal if they are perpendicular". Would you be satisfied with "two lines are orthogonal if their vectors are perpendicular"? --beefyt (talk) 21:28, 30 January 2009 (UTC)
replies: text: Why not "two lines are orthogonal if they meet at a right angle"? Michael Hardy (talk) 22:17, 30 January 2009 (UTC)
replies: text: The direction of a line in a two-dimensional plane is defined by a two-dimensionsl vector. Also, the direction of a line in three-dimensional space is defined by a three-dimensional vector. Two lines are perpendicular if and only if their defining vectors are perpendicular to each other. This is taught in advanced high-school mathematics. 98.67.108.12 (talk) 00:22, 25 August 2012 (UTC)
thread_title:  Intro Rewrite 

message: Isn't orthogonal a T shape? Can it say so on the article as a better description than right angle? Or, if the orthogonal is the L shape, is it OK to say "An Orthogonal is an L shaped intersection." and provide a nice L shaped joint picture? I can't remember if it is T or L and I cannot read complex formulae. ~ R.T.G 15:21, 30 January 2009 (UTC)
replies: text: Two lines are orthogonal if they meet at a right angle.  Thus the strokes of L are orthogonal, as are those of T or +. —Tamfang (talk) 03:08, 23 October 2011 (UTC)
thread_title:  T - shape? 

message: You might be interested that the book Applied Mathematics for Database Professionals refers readers to this Wikipedia article. - 114.76.235.170 (talk) 14:17, 23 June 2010 (UTC)
thread_title:  Interesting 

message: Why, what, where, and how does this article need additional citations for verification? Hyacinth (talk) 05:26, 2 August 2010 (UTC)
replies: text: Where it says "citation needed." --KSnortum (talk) 18:06, 13 January 2012 (UTC)
thread_title: Additional citations

message: Is there a good reason for (a, g, and n) to be enclosed in ()? —Tamfang (talk) 03:07, 23 October 2011 (UTC)
thread_title:  (''a'', ''g'', and ''n'') 

message: Orthogonality vs. Independence in random variables and statistics.
Independence is a much stronger specification or assumption than orthogonality or uncorrelated.
Orthogonal means that E(XY) = 0.
If E(XY) = E(X)E(Y), then X and Y are uncorrelated.
The above must not be confused by the following.
If two random variables or statistics X and Y are jointly Gaussian;
And X and Y are both zero mean; [This is often forgotten about.]
And X and Y are orthogonal;
Then X and Y are independent.
Otherwise, the independence of X and Y has to be considered on a case-by-case basis.
For independence, if f(x,y) is the joint probability density of X and Y, then we must have f(x,y) = f(x)f(y). There is no other way.
If E(XY) = 0, and either E(X) = 0 or E(Y) = 0, then X and Y are uncorrelated because 
E(XY) = E(X)E(Y) = 0.  — Preceding unsigned comment added by 98.67.108.12 (talk) 00:50, 25 August 2012 (UTC)
thread_title:  Orthogonality vs. Independence in .... 

message: Attributing the concept of orthogonality (really perpendicularity) to the Babylonians or the Egyptians or whatever is probably not reasonable.  All ancient mathematical civilizations (Babylonian, Egyptian, Indian, Chinese) had the concept of perpendicularity in two dimensions -- a discussion which belongs in perpendicularity.  I am not sure that any of them had any of the generalizations that are called "orthogonality". --Macrakis (talk) 00:51, 25 January 2013 (UTC)
thread_title: Ancient history

message: The bottom of this page is lacking in succinctness, if you ask me. Many if not all of the smaller subsections starting at "Art and Architecture" should be grouped into a single "Definitions in other fields" section. The "Statistics, econometrics, and economics" section mentions nothing about economics and the only comments on econometrics can be generalized to optimization problems overall (and are only tangentially related to orthogonality, questioning the importance in an article devoted to explaining the concept of orthogonality). Furthermore, the general background knowledge assumed in the reader is very different for each heading. I propose the section be condensed with easy to comprehend definitions, more like the Taxonomy section (which defines its jargon somewhat) than the Neuroscience section (which leaves its jargon to the imagination without even linking another page or reference).
 
Also, rather than bring up individual tidbits that relate to or require orthogonality (I'm looking at econometrics again...), maybe a dedicated section called "Properties that follow from orthogonality" might be useful. Compiling this list would be difficult for one person, but if we could bring back all the people that made those ridiculous subheadings, they'd be able to fill a good amount of it in.  — Preceding unsigned comment added by 2602:304:AB31:95F9:DD98:CE54:FF67:41C6 (talk) 17:36, 31 May 2013 (UTC)
thread_title: Alternate Definitions

message: Under etymology it is stated that the Greek term ορθογώνιον (= orthogonal) was used to denote a rectangle, and then it came to mean "right triangle." This is incorrect.
A right triangle is denoted by "ορθογώνιον τρίγωνον" which literally translates to "right angle triangle." The term "ορθογώνιον" without being accompanied by the word "τρίγωνον" (= triangle) is still used to denote a rectangle, and was never used differently.
Note: Not sure whether the Latin term "orthogonalis" came to mean "right triangle" or not; I'm only certain about the Greek term.
62.195.45.181 (talk) 15:04, 1 May 2015 (UTC)
thread_title:  Problem with etymology section 

message: Orthogonality covers both perpendicularity and hyperbolic orthogonality. The former occurs with a definite quadratic form, the latter with an isotropic quadratic form. The lead section included mention of logical independence, uncorrelated variables, and higher dimensions, all distractions from a planar phenomena widely applied. With 189 watchers, 24 reviewing recent changes (people are out-of-town for summer holidays), editors may be interested in improving this article before September. This is a technical article with well-known concepts, so efforts to obscure the meaning will stand out. — Rgdboer (talk) 22:02, 6 August 2017 (UTC)
replies: text: The issue here is WP:UNDUE. I am not saying that this material doesn't belong in the article, just that it doesn't belong in the lead, as it has only one or two sentences devoted to it in the body of the article. Orthogonality is used in many ways in mathematics, not all of which need be mentioned in the lead. For instance, the orthogonality relationship of Latin squares, which does not fit into your two types classification, and I would not mention in the lead. The lead should give a reader the chance to decide if the article is worth investing time in and slinging around a bunch of obscure terms and applications does not help that goal.--Bill Cherowitzo  (talk) 23:15, 6 August 2017 (UTC)
thread_title:  Two types