Wikipedia

Talk:Quaternion: Difference between revisions

Browse history interactively
← Previous edit
Content deleted Content added
VisualWikitext
Seub (talk | contribs)
52 edits
No edit summary
 
(214 intermediate revisions by 61 users not shown)
Line 1:
{{WikiProject banner shell |class=B |1=
{{maths rating|frequentlyviewed=yes|importance=Mid|field=algebra|class=C}}
{{WikiProject Mathematics}}
}}
{{OnThisDay |date1=2006-10-16|oldid1=81116033|date2=2007-10-16|oldid2=164506304|date3=2008-10-16|oldid3=245235137|date4=2009-10-16|oldid4=320120321|date5=2010-10-16|oldid5=391001253}}
{{Daily pageviews}}
{{User:MiszaBot/config
|maxarchivesize = 80K150K
|counter = 4
|minthreadsleft = 510
|minthreadstoarchive = 1
|algo = old(90d365d)
|archive = Talk:Quaternion/Archive %(counter)d
}}
{{Archive box |search=yes |bot=Lowercase sigmabot III |age=12 |units=months |auto=long }}
{{OnThisDay |date1=2006-10-16|oldid1=81116033|date2=2007-10-16|oldid2=164506304|date3=2008-10-16|oldid3=245235137|date4=2009-10-16|oldid4=320120321|date5=2010-10-16|oldid5=391001253}}
 
{{archive box |auto=long |bot=MiszaBot I |age=90 }}
== Overly technical ==
 
The article is currently too technical for non-experts to understand; I am adding a tag to suggest the article be improved to be understandable to non-experts. [[User:Betanote4|Betanote4]] ([[User talk:Betanote4|talk]]) 18:16, 5 August 2020 (UTC)
 
:A certain amount of it is accessible to non-experts, and a certain amount of it isn't. But it's a technical subject, and that's about what we should expect. Unless you can be more specific, the tag isn't really helpful, so I've removed it. –[[User:Deacon Vorbis|Deacon Vorbis]] ([[User Talk:Deacon Vorbis|carbon]] • [[Special:Contributions/Deacon Vorbis|videos]]) 18:30, 5 August 2020 (UTC)
 
:This article is "Quaternions for Pure Mathematicians". For practical aspects, see [[Quaternions and spatial rotations]], with [[Quaternions and spatial rotations#Quaternions]] providing a quick introduction to quaternions. [[User:BMJ-pdx|BMJ-pdx]] ([[User talk:BMJ-pdx|talk]]) 22:55, 5 June 2023 (UTC)
== regarding basis shown in ''Matrix representations'' ==
 
== Quaternions are four-dimensional ==
Even though the section says that there are at least two ways, should'nt it be explicitly said that the basis made up of four 4x4 matrices shown in the example are ''not'' unique and that other matrices which have the same properties can be used to represent i,j and k. '''also how many such bases can be possible?''' a trivial case is a basis which is made of the transpose (equivalent to choosing a basis of -i, -j and -k) or basis where matrices corresponding to i, j and k are cyclicaly shifted. does another basis which cannot be made up by doing these two operations exist? Does the basis have to be made up of 0, 1 and -1?
[[User:Cplusplusboy|Cplusplusboy]] ([[User talk:Cplusplusboy|talk]]) 13:22, 20 January 2012 (UTC)
 
We're missing h, it's call the Hamilton set for a reason, Hamilton was a human...
:These questions make decent research projects, but they will not be appropriate for the article (unless there is some very nice ''citable'' result). (Ordered) bases of the type you described will correspond naturally to the ring automorphisms of '''H'''. [[User:Rschwieb|Rschwieb]] ([[User talk:Rschwieb|talk]]) 13:58, 20 January 2012 (UTC)
 
<math display=block>a\ w\ \mathbf h + b\ x\ \mathbf i + c\ y\ \mathbf j +d\ z\ \mathbf k,</math> [[User:Nbritton|NJB]] ([[User talk:Nbritton|talk]]) 04:52, 6 January 2023 (UTC)
:: Arbitrary 4 × 4 real matrix [[Jordan normal form|without Jordan blocks]] with same [[eigenvalue]]s (namely, {{{mvar|i}}, {{mvar|i}}, −{{mvar|i}}, −{{mvar|i}}} ) is eligible to [[Algebra representation|represent]] the quaternion {{mvar|i}}. You may construct real 4-dimensional quaternions' representations by algebraic conjugation: {{math|1={{mvar|X}} → {{mvar|U}}<sup>−1</sup> {{mvar|X}} {{mvar|U}}}} where {{mvar|X}} is a canonical representation and {{mvar|U}} is an arbitrary reversible 4 × 4 real matrix chosen for this particular representation. This is actually nothing but a (two-side) [[intertwiner]], or simply a [[change of basis]], and is considered the same in the [[representation theory]]. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 14:30, 20 January 2012 (UTC)
::<small>(Note to OP: the conjugation described here produces an automorphism of '''H'''. [[User:Rschwieb|Rschwieb]] ([[User talk:Rschwieb|talk]]) 15:12, 20 January 2012 (UTC)</small>
::: <small>It is an automorphism of ℍ only if {{mvar|U}} belongs to [[SO(4)]]. I guess that it is also sufficient (the 3-sphere of unit quaternions in the canonical representation seems to be the same as left-isoclinic rotations), but am not completely sure. Moreover, as we discuss representations by ''arbitrary'' matrices, {{mvar|U}} does not even have to be orthogonal, this means that {{mvar|U}}<sup>−1</sup> {{mvar|X}} {{mvar|U}} not necessary is a canonical representation of ''any'' quaternion. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 16:20, 20 January 2012 (UTC)</small>
::::<small>Oh. I've never heard of a reversible matrix, so I was guessing it meant special orthogonal. [[User:Rschwieb|Rschwieb]] ([[User talk:Rschwieb|talk]]) 20:25, 20 January 2012 (UTC)</small>
:::::<small>Having considered the group of matrices that may be ''U'', this does not directly say the obvious things about the resulting representation. For example, the first matrix always remains the identity matrix. Next, it would seem to me that the remainder of the basis matrices obey a linear transformation law, which, unlike ''U'', has only three dimensions: the symmetry group of S<sup>2</sup>? — [[User:Quondum|Quondum]][[User_talk:Quondum|<sup>☏</sup>]][[Special:Contributions/Quondum|✎]] 05:18, 21 January 2012 (UTC)</small>
 
: It’s hard to tell what you are trying to say. But as a general rule, Wikipedia follows whatever the commonly accepted convention is in reliable sources (or when there are multiple common convention, picks one and mentions the alternatives). Do you have a reliable source for your "h" here? If you are just throwing out ideas, you may want to write a blog post or self-published paper, as you are unlikely to find support for their inclusion here. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 07:25, 6 January 2023 (UTC)
:::Ahem. Perhaps we can keep this to language accessible to those who do not already know the answer to the original question? Cplusplusboy may have a point that "There are at least two ways of representing quaternions as matrices" may be so weak a statement as to be misleading, and should at least be rephrased. There are an infinite number of ways (for example derivable from each of those representations via 3-dimensional rotations and reflections of the (''i'',''j'',''k'') basis on a 4×4 real matrix representation alone (the ring automorphism group being isomorphic to [[O(3)]], I guess). So perhaps it would be reasonable to change this to "There are many ways of representing quaternions as matrices" – even without citations. Those given just happen to be two of the "neat" ways. — [[User:Quondum|Quondum]][[User_talk:Quondum|<sup>☏</sup>]][[Special:Contributions/Quondum|✎]] 14:50, 20 January 2012 (UTC)
::::Hehe, I'm not very familiar in this math and so didn't want to edit the article myself. I was just comparing an example given in a book on quaternions and found that the bases it showed differed from wikipedia's. Since I was under the impression that the basis was unique, I tried to do the check of ijk=-1 property on both bases and found that both were right and wanted to confirm the fact. Should this talk be removed as the confusion is resolved? I didn't see anything about that in the guidelines. [[User:Cplusplusboy|Cplusplusboy]] ([[User talk:Cplusplusboy|talk]]) 16:36, 21 January 2012 (UTC)
:::::I've edited the article in an attempt to address the initial problem; we'll see what others make of it. No, we leave the discussion as is; there are tight constraints on any editing of prior comments; it'll be removed in due course by the archiving process. See [[Wikipedia:Talk page guidelines#Editing comments]]. — [[User:Quondum|Quondum]][[User_talk:Quondum|<sup>☏</sup>]][[Special:Contributions/Quondum|✎]] 06:35, 22 January 2012 (UTC)
 
:: It's basic primary school mathematics, you shouldn't need a source for <math>h</math>, as it is the original human number system otherwise known as <math>\mathbb{R}</math>. It's always there, but it's most often omitted due to a combination of short hand mathematical notation and/or ignorance about [https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF lateral numbers].
Please, in an article on mathematics, be more precise. For instance: The sentence ,Using 4x4 real matrices ...' is clear, since a,b,c,d must be real numbers, but this should be stated there, even if this is tedious. But in the 2-dimensional matrix representation some lines above, nothing is clear: Are the a,b,c,d real numbers as well or complex numbers? Obviously complex!? And why there are different letters for a,b,c,d in the text and in the matrix representation? And the same for i. Is this the same complex unit as in the text line before. The same question some blocks before in the determinant - please state whether these a,b,c,d are real or complex numbers. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/130.133.155.70|130.133.155.70]] ([[User talk:130.133.155.70|talk]]) 13:17, 18 July 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
 
:: To add two signed numbers, such as <math>+1 + -1</math>, most people would just rewrite the equation using subtraction and say that the value is <math>0</math>, but this because they were taught short hand arithmetic notation starting in primary school. Even when dealing with only real numbers, 0i is always still there as part of the equation, it's also just simply omitted in short hand notation.
== Error? ==
 
::<math>+1 + -1 = (+1 + 0i) + (-1 + 0i)) = 0 + 0i</math>
I've never made an edit (except for the occasional spelling fix) so not sure of protocol.
 
:: So naturally in the equation above, the <math>0</math> alone by itself is on the <math>h</math> axis, <math>\mathbb{R}</math>, which is omitted in short hand notation, but the more formal general form for this equation using quaternions is as follows:
The section "Three-dimensional and four-dimensional rotation groups" refers to the 3-sphere as a three dimensional sphere, it isn't, the 3-sphere is four dimensional (its hypersurface has 3 dimensions)
 
::<math>(1h + 0i + 0j + 0k) + (-1h + 0i + 0j + 0k) = 0h + 0i + 0j + 0k = 0</math>
[[User:Ds1392|Ds1392]] ([[User talk:Ds1392|talk]]) 14:25, 20 October 2013 (UTC)
:The [[3-sphere]] or sphere of dimension three is a [[manifold]] of dimension 3 that may be [[embedding|embedded]] as an [[hypersurface]] in the [[Euclidean space]] of dimension 4. This embedding is realized by defining the 3-sphere as the zero set of the equation <math>x^2+y^2+z^2+t^2-1=0.</math> Thus the article is correct, although somehow too technical.
:There is no protocol for editing. You have just to edit. However, if your edit is wrong or does not follows Wikipedia rules and policies, it is likely that it will be quickly reverted. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:48, 20 October 2013 (UTC)
 
::Point taken :D Perhaps the wording could be adjusted a smidge to make that clearer? I guess it's difficult to satisfy both the requirement that wikipedia be readable by a general audience (where intuitively, an n-dimensional object is one that can be embedded in '''R'''<sup>n</sup>) and the requirement that the information be accurate (an n sphere is an n-dimensional manifold.) If I can think of a way to improve the phrasing that isn't too wordy, I'll make the edit. [[User:Ds1392|Ds1392]] ([[User talk:Ds1392|talk]]) 01:19, 21 October 2013 (UTC)
 
:::<math display=inline>1 = h^2i^2j^2k^2 = (\sqrt {-1})^2(\sqrt {-1})^2(\sqrt {-1})^2(\sqrt {-1})^2</math>
::: I don't think it can be clarified without properly distinguishing what "dimensions" are being discussed. It has (geometric) dimension 4 when embedded in R<sup>4</sup>, but has (topological) dimension 3. [[User:Rschwieb|Rschwieb]] ([[User talk:Rschwieb|talk]]) 13:23, 21 October 2013 (UTC)
 
::::No, an ''n''-sphere always has dimension ''n''. It can be embedded in a larger dimensional space, but that does not change its dimension. See [[manifold]]. [[User:Ozob|Ozob]] ([[User talk:Ozob|talk]]) 14:07, 21 October 2013 (UTC)
:::::I agree with Ozob. The problem may come that for many people the distinction between a sphere and a ball is unclear: The sphere of dimension ''n'' is the boundary of the ball of dimension ''n''+1. The surface of Earth is roughly a 2-sphere, while Earth in the whole is roughly a 3-ball. The lead of [[Sphere]] deserve to be edited to emphasize this distinction. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:28, 21 October 2013 (UTC)
::::::@Ozob (cc @D.Lazard): I'm saying that there are two subsets of humans: those who think of dimension in the topological way and those thinking of it in the geometric way. I'm pretty sure most laypeople carry the geometric dimension learned in grade school through 2-d an 3-d geometry. So, for example, they will report that the 2-sphere is a "three dimensional object," even if it is just the surface of a ball. I've seen this misconception cleared up a handful of times in graduate and undergraduate setting, so it is even common among non-laypersons.
:::::: Anyhow in summary, ''you and I know'' it has an intrinsic dimension that doesn't depend on where it's embedded, but stubbornly pretending that everyone else will understand it that way if we say so is an invitation for misunderstanding. [[User:Rschwieb|Rschwieb]] ([[User talk:Rschwieb|talk]]) 15:00, 21 October 2013 (UTC)
:::::::Nobody would say that a line is anything more than one dimensional or that a plane is anything more than two dimensional. I agree that some people are confused about the precise meaning of dimension, but the standard definition is both not too surprising and used universally within mathematics. I don't see any reason why this article should equivocate on this point. [[User:Ozob|Ozob]] ([[User talk:Ozob|talk]]) 18:44, 21 October 2013 (UTC)
::::::::I've changed it to "[[3-sphere]] S<sup>3</sup>". In this instance, using a ''less'' familiar term might lead to less confusion, for the reason that it does not as readily trigger an unintended interpretation. — [[User_talk:Quondum|''Quondum'']] 00:41, 22 October 2013 (UTC)
:::::::::This change works for me. I was tripped up even though I ''should'' know better. Quondum's point re "less familiar" terms is quite valid; It's probably a good idea to avoid phrases with a natural language interpretation as much as possible because it's hard to avoid the reflexive interpretation. In everyday speech I refer to the 3-ball/2-sphere-in-'''R'''<sup>3</sup> as a "three dimensional sphere" (formally correct or not this is how natural language is, and natural language "got there first" so to speak.) If I'm referring to the manifold I'll explicitly use the term "3-sphere" to avoid ambiguity. My 2c anyway :) [[User:Ds1392|Ds1392]] ([[User talk:Ds1392|talk]]) 12:13, 23 October 2013 (UTC)
 
:::<math display=inline>h^2 = \frac {1}{i^2j^2k^2} = \frac {1}{(-1)(-1)(-1)} = -1</math>
== Matrix vector product ==
:::<math display=inline>i^2 = \frac {1}{h^2j^2k^2} = \frac {1}{(-1)(-1)(-1)} = -1</math>
:::<math display=inline>j^2 = \frac {1}{h^2i^2k^2} = \frac {1}{(-1)(-1)(-1)} = -1 </math>
:::<math display=inline>k^2 = \frac {1}{h^2i^2j^2} = \frac {1}{(-1)(-1)(-1)} = -1</math>
 
The following text was removed from Matrix representation:
:The multiplication of two quaternions
: <math>
ab = c
</math>
:can be represented by matrix vector multiplication:
: <math>
\begin{bmatrix}
a_0 & -a_1 & -a_2 & -a_3 \\
a_1 & a_0 & -a_3 & a_2 \\
a_2 & a_3 & a_0 & -a_1 \\
a_3 & -a_2 & a_1 & a_0
\end{bmatrix}
\begin{bmatrix}
b_0 \\ b_1 \\ b_2 \\ b_3
\end{bmatrix}
:=
\begin{bmatrix}
c_0 \\ c_1 \\ c_2 \\ c_3
\end{bmatrix}
</math>
:If we define
:<math>
B(a) \equiv
\begin{bmatrix}
a_0 & -a_1 & -a_2 & -a_3 \\
a_1 & a_0 & -a_3 & a_2 \\
a_2 & a_3 & a_0 & -a_1 \\
a_3 & -a_2 & a_1 & a_0
\end{bmatrix}
</math>
:and ''a'', ''b'', and ''c'' are real column vectors constructed from quaternions, we can rewrite the multiplication as
: <math> B(a) b = c </math>
:or
: <math> B(a) B(b) = B(c) </math>.
:We can also define
: <math>
A(a) \equiv
\begin{bmatrix}
a_0 & a_1 & a_2 & a_3 \\
-a_1 & a_0 & -a_3 & a_2 \\
-a_2 & a_3 & a_0 & -a_1 \\
-a_3 & -a_2 & a_1 & a_0
\end{bmatrix}
</math>.
:The A and B matrix constructions have the following basic properties.
: <math> A(a) b = B(b) a^* </math>
: <math> A(a) B(b) = B(b) A(a) </math>
 
Two matrices must be multiplied to represent the quaternion product. The text removed today was unreferenced and made a false assertion.::[[User:RgdboerNbritton|RgdboerNJB]] ([[User talk:RgdboerNbritton|talk]]) 2016:51, 186 JulyJanuary 20142023 (UTC)
 
:::Regardless of all that stuff, you still need a reliable source, and [[WP:YOUTUBE|YouTube]] is not a reliable source.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 17:04, 6 January 2023 (UTC)
:While it would be easy enough to correct this, the matrix representations as they stand in the article are sufficient. Also not being referenced makes it look like the OR it probably is. I agree with the removal. —[[User_talk:Quondum|Quondum]] 21:28, 18 July 2014 (UTC)
:::(a) This Youtube video is well made and worth showing to students but it does not support your claims.
:::(b) What you have written here is not the way people use quaternions or other number systems in practice. Again, feel free to invent whatever number system you want in your own writings (self-published papers, blog posts, etc.). However, it’s not relevant to this Wikipedia article. Let’s try to keep discussion focused on improving the article; Wikipedia talk pages are not a general-purpose forum. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:03, 6 January 2023 (UTC)
::::In his notation, <math>(1h + 0i + 0j + 0k) \times (1h + 0i + 0j + 0k) = -1h + 0i + 0j + 0k </math>. That is not how multiplication in quaternions works.--[[User:ArnoldReinhold|agr]] ([[User talk:ArnoldReinhold|talk]]) 18:47, 6 January 2023 (UTC)
:::::I was not attempting to multiple them, it was just simple addition on a system of polynomial equations, here is a better example..
:::::<math>+a(x-y)^2h^2 + 0(x-y)^2i^2 + 0(x-y)^2j^2 + 0(x-y)^2k^2</math>
:::::<math>-a(x-y)^2h^2 + 0(x-y)^2i^2 + 0(x-y)^2j^2 + 0(x-y)^2k^2</math>
:::::There is an implied symbolic coefficient, in this instance the coefficient of <math>i</math>, <math>j</math>, and <math>k</math> is <math>0</math>, so when multiplied those all evaluate to <math>0</math>.
:::::<math>+a(x-y)^2h^2</math>
:::::<math>-a(x-y)^2h^2</math>
:::::The only difference here is <math>a + -a</math>, so you can use addition here. [[User:Nbritton|NJB]] ([[User talk:Nbritton|talk]]) 21:33, 6 January 2023 (UTC)
::::::Your expressions here are incoherent because you have not defined a, x, y, or h. But don’t bother to define them; it’s a waste of your (and everyone’s) time. If you have a question (Example question: {{tq|i=yes|"why when writing complex numbers don't mathematicians give a symbolic name to the real unit {{math|1=''h'' ≡ 1}}, so a complex number could be written {{math|''xh'' + ''yi''}} with real part {{mvar|x}} and imaginary part {{mvar|y}}, so there would be symbolic symmetry between real and imaginary parts?"}} Example answer: {{tq|i=yes|"You are welcome to do it that way if you want. Most mathematicians like to skip redundant symbols where possible to reduce clutter."}}) perhaps take it to [[Wikipedia:Reference desk/Mathematics]] or try some mathematics discussion forum like reddit or mathematics stack exchange. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 22:30, 6 January 2023 (UTC)
:::::::Why? a, x, and y have no relevance to the topic at hand, we are talking about our four-demential quaternion number system, the entirety of the equations including the Arabic numerals themself are symbolic representations of abstract ideas. We're dealing with algebra here, and anyone who has passed college algebra knows that you can have coefficients and variables in an equation.
:::::::What you call clutter I call lack of completeness and attention to detail, you've just literally admitted the point that I was originally attempting to make, that the wikipedia article is using sloppy shorthand notation in its attempt to rigorously define what a quaternion even is. If you don't see that as a problem then I don't know what to tell you. I have no problem with people using shorthand notation, I do it all the time, but in an encyclopedic article that is attempting to rigorously define something, you have to at least discuss the formal long form notation at least one, and ideally also provide wiki links to all of the assumed knowledge that is necessary to understand it. [[User:Nbritton|NJB]] ([[User talk:Nbritton|talk]]) 20:05, 7 January 2023 (UTC)
::::::::Wikipedia is using the standard notation used by literally every source about this topic for the past 150 years. You can run a weird crusade against conventions you dislike somewhere else, but the job of Wikipedia is to describe encyclopedic topics as established in reliable sources, which is what this article currently does.
::::::::Pretty much anyone who does mathematics can think of several notation conventions they dislike for one reason or another, but if they want to change the conventions they can make those arguments in blog posts, journal papers, textbooks, etc.; speculative conversations about possible non-standard notation conventions don’t belong in wikipedia.
::::::::{{tq|i=yes|"I don't know what to tell you."}} – that’s fine, you don't need to tell us anything. This whole conversation is off topic and should end ASAP. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 21:35, 7 January 2023 (UTC)
 
== Biquaternions are eight-dimensional ==
::But this looks very useful. Why would you multiply the whole matrix; that's four times the amount of work???
 
See [[biquaternion]] for the use of h as a square root of minus one which commutes with i, j, and k. The algebra of biquaternions is only four dimensions when considered over the field of complex numbers x + yh. Biquaternions provide a representation of Minkowski space and Lorentz transformations described by [[Ludwik Silberstein]] in 1914, but the original algebra comes from Hamilton's ''Lectures on Quaternions'' (1853). [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 03:32, 8 January 2023 (UTC)
::Think it was just transposed incorrectly:
 
== Why refer to i, j, and k as “basic quaternion”? ==
:: <math>
\begin{bmatrix}
a_0 & a_1 & a_2 & a_3 \\
-a_1 & a_0 & -a_3 & a_2 \\
-a_2 & a_3 & a_0 & -a_1 \\
-a_3 & -a_2 & a_1 & a_0
\end{bmatrix}
\begin{bmatrix}
b_0 & b_1 & b_2 & b_3 \\
-b_1 & b_0 & -b_3 & b_2 \\
-b_2 & b_3 & b_0 & -b_1 \\
-b_3 & -b_2 & b_1 & b_0
\end{bmatrix}
:=
\begin{bmatrix}
c_0 & c_1 & c_2 & c_3 \\
-c_1 & c_0 & -c_3 & c_2 \\
-c_2 & c_3 & c_0 & -c_1 \\
-c_3 & -c_2 & c_1 & c_0
\end{bmatrix}
</math>
 
“ Quaternions are generally represented in the form
:: <math>
a + bi + cj + dk
\begin{bmatrix}
where a, b, c, and d are real numbers; and i, j, and k are the basic quaternions.”
a_0 & a_1 & a_2 & a_3
\end{bmatrix}
\begin{bmatrix}
b_0 & b_1 & b_2 & b_3 \\
-b_1 & b_0 & -b_3 & b_2 \\
-b_2 & b_3 & b_0 & -b_1 \\
-b_3 & -b_2 & b_1 & b_0
\end{bmatrix}
:=
\begin{bmatrix}
c_0 & c_1 & c_2 & c_3
\end{bmatrix}
</math>
 
Why refer to i, j, and k as the “basic quaternions” and not the “standard basis vectors”? I have not seen the term basic quaternion before and did not find any relevant information when looking it up. [[Special:Contributions/76.151.136.63|76.151.136.63]] ([[User talk:76.151.136.63|talk]]) 13:40, 26 January 2023 (UTC)
:: <math>
\begin{bmatrix}
a_0 & -a_1 & -a_2 & -a_3 \\
a_1 & a_0 & a_3 & -a_2 \\
a_2 & -a_3 & a_0 & a_1 \\
a_3 & a_2 & -a_1 & a_0
\end{bmatrix}
\begin{bmatrix}
b_0 \\ b_1 \\ b_2 \\ b_3
\end{bmatrix}
:=
\begin{bmatrix}
c_0 \\ c_1 \\ c_2 \\ c_3
\end{bmatrix}
</math>
 
:Firstly, the standard basis of the vector space of the quaternions contains also the real number 1. Secondly, one may understand what are quaternions without knowing vector spaces, bases of vector spaces, and standard bases. So, the change you suggest would make the article unnecessarily more technical (see [[WP:TECHNICAL]]). Also, the concept of a vector space has been introduced years after the quaternions, and I guess that bases of vector spaces have been so named after the ''basic'' quaternions. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:35, 26 January 2023 (UTC)
::[[Special:Contributions/213.205.240.129|213.205.240.129]] ([[User talk:213.205.240.129|talk]]) 13:06, 12 September 2014 (UTC)
 
::I changed ''basic quaternions'' to ''basis vectors'' or ''basis elements'' partly to be consistent with the rest of the article, partly because I found a reference for it, and partly because ''basic quaternions'' seems to be nonstandard.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 17:04, 26 January 2023 (UTC)
== Type of isomorphism is unclear ==
:::It's a matter of changing definitions of ''vector''. The use of the word ''vector'' in mathematics was originated by Hamilton to refer to the "imaginary" part of a quaternion. But later Gibbs/Heaviside adopted it in their formulation of electrodynamics based on dot and cross products (popularized in the book ''[[Vector Analysis]]''). Later, while physicists continue to use the Gibbs/Heaviside concept, mathematicians adopted the same name for the broader concept of a [[vector space]]. The mathematician's concept of a "vector" is different enough that applying the word to the imaginary part of a quaternion causes some confusion today. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 17:22, 26 January 2023 (UTC)
 
::::I don't see this in the History section, so perhaps it should be included.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 17:42, 26 January 2023 (UTC)
In the section [[Quaternion#Matrix_representations]], the following sentence occurs:
:::::There is an additional point of confusion, which is that as the even sub-algebra of the [[geometric algebra]] (real [[Clifford algebra]]) of Euclidean 3-space, the quaternions are "actually" made up of a scalar ("real") part and a ''[[bivector]]'' ("imaginary") part. Both Hamilton and Gibbs/Heaviside somewhat conflated the concepts of vectors (line-oriented magnitudes) and bivectors (plane-oriented magnitudes), sometimes calling the latter "[[pseudovector]]s" or "axial vectors" because they transform differently than ordinary vectors, the "polar vectors". This is possible in 3 dimensional Euclidean space (but no other dimension) because every plane has a unique perpendicular axis. When you take the [[cross product]] of two vectors to get a "pseudovector", it would be conceptually clearer to instead take the [[wedge product]] of two vectors to get a bivector, treated as a conceptually different type of object. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 20:11, 26 January 2023 (UTC)
*Restricted to unit quaternions, this representation provides an [[group isomorphism|isomorphism]] between [[3-sphere|''S''<sup>3</sup>]] and [[SU(2)]].
This sentence has the problem that technically it is ill-defined, or more correctly, since both these objects are only in the same category as sets, this only says that they have the same cardinality. I expect that most people will find a natural interpretation as an isomorphism of topologies and/or as a congruence of geometric objects in Euclidean 4-space. Given that the representation is given as the basis of the isomorphism, the geometric interpretation may be intended, but is inappropriate (we would not normally call a linear mapping between representations an isomorphism in the algebraic context). However, ''S''<sup>3</sup> regarded as a topological object or a geometric object, '''H''' regarded as a ring and [[SU(2)]] regarded as a group leaves room for confusion of what isomorphism is meant. Could someone with more knowledge in the area please qualify this to clarify what is meant? Perhaps leave ''S''<sup>3</sup> out of it altogether, and simply state that there is a group isomorphism between the multiplicative group of unit quaternions and SU(2)? —[[User_talk:Quondum|Quondum]] 13:56, 19 July 2014 (UTC)
 
== Square roots of arbitrary quaternions ==
:Where to even begin.... there's lots of confusion all around, it should be clarified.
:* (lower-case) [[su(2)]] is an [[algebra]], and specifically a [[Lie algebra]].
:* the quaternions are an algebra, too. (an algebra is a vector space endowed with multiplication, usually a non-commutative multiplication)
:* the [[structure constant]]s of su(2) are equal to those of H, except that they are multiplied by an extra factor of <math>\sqrt{-1}</math>. Thus, the generators of su(2) when squared, give you +1, instead of -1 when you square the generators of H.
 
The formula for the square root of a quaternion essentially uses the trigonometric identity for the sine of a half angle $\sin(\theta/2) = \sqrt{(1-\cos(\theta))/2}$. The formula looses precision for small angles and should never be used for numerical calculation. This is similar to finding the angle between two vectors using arccos formula, which is generally unacceptable. [[User:Arcshinus|Arcshinus]] ([[User talk:Arcshinus|talk]]) 02:50, 15 March 2023 (UTC)
:* (upper case) SU(2) is a [[Lie group]] it corresponds to the [[fundamental representation]] of the algebra. Give a point <math>\vec{\theta}</math> in the lie algebra su(2), you get the corresponding group element <math>U=exp(i\vec{\theta}\cdot\vec{\sigma})</math>, which is a 2x2 unitary matrix. Here exp is the [[exponential map]] used to convert dirction vectors into [[geodesics]]. The <math>\vec{\sigma}</math> are the [[Pauli matricies]].
 
== Discovery or invention? ==
:* You can do exactly the same thing as above, using +1, i,j,k instead of using the identity matrix plus the pauli matricies. You get exactly the same thing (except for an extra confusing factor of <math>\sqrt{-1}</math> that floats around and makes thing randomly confusing.
 
To me, it seems that some things in mathematics are discoveries, and some are inventions. I consider <math>pi</math> and <math>e</math> to be discoveries, since they are fundamental to so much. Matrices I consider to be an invention, since, despite their flexibility and utility value, I've always regarded them as being rather arbitrary (full disclosure: I never did like matrices :). Quaternions also seem to fall into the invention category (more full disclosure: I love quaternions). Complex numbers are harder to so categorize; while the term "imaginary part" may argue for "invention", they are so closely tied to fundamentals (e.g., two-dimensional Euclidean space) that "discovery" also seems accurate. [[User:BMJ-pdx|BMJ-pdx]] ([[User talk:BMJ-pdx|talk]]) 22:39, 5 June 2023 (UTC)
:* The 3x3 matrix group representation of su(2) is call the [[rotation group]] [[SO(3)]]. The explicit mapping is this. Let <math>\vec{v}</math> be a 3D vector. Let R be a 3x3 rotation matrix. Then, <math>R\cdot\vec{v}=U^\dagger \vec{v} \cdot\vec{\sigma} U</math> where U is same as above. By contrast, R is given by <math>exp(\vec{L}\cdot\vec{\theta}/2)</math> where <math>\vec{L}</math> are the generators of [[angular momentum]] i.e. the purely real 3x3 matrixes that generate [[SO(3)]] rotations. anyway, its the same theta in U and R.
 
:Maybe make a blog or social media post out of this instead of chitchatting about it here. Cf. [[WP:NOTFORUM]]. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 00:46, 6 June 2023 (UTC)
:* (upper case) SU(2) is a manifold that is topologically isomorphic to S_3
:See [[Philosophy of Mathematics]]. --[[Special:Contributions/50.47.155.64|50.47.155.64]] ([[User talk:50.47.155.64|talk]]) 15:51, 15 August 2023 (UTC)
::Many quantum and particle physicists would say that hypercomplex numbers fully exist in the natural world. [[User:LagrangianFox|LagrangianFox]] ([[User talk:LagrangianFox|talk]]) 18:58, 11 October 2024 (UTC)
 
== Error in the introduction? ==
:* (upper case) SO(3) is a topological manifold that is [[covering|covered twice over]] ([[double cover]]) by SU(2).
 
I think the sentence: 'Quaternions are generally represented in the form where the coefficients a, b, c, d are real numbers, and 1, i, j, k are the basis vectors or basis elements.'
:* The [[complex projective plane]] CP(2) is topologically isomorphics to SU(2)
 
Should read: 'Quaternions are generally represented in the form where the coefficients a, b, c, d are real numbers, and i, j, k are the basis vectors or basis elements.'
:* The last bit is made use of in quantum mechaics, where a [[spinor]] is a 2D complex vector of unit length (thus its projective) and is spun around by elements of SU(2).
 
That is, the '1, ' before the 'i' should be deleted. Is that correct? [[User:MathewMunro|MathewMunro]] ([[User talk:MathewMunro|talk]]) 09:09, 15 February 2024 (UTC)
:* The resulting manifold is called the [[Bloch sphere]].
 
:The introduction is correct, quaternions form a [[vector space]] of dimension 4 over the reals. However, the modern concept of a [[vector space]] was not elaborated when Hamilton introduced quaternions, and this may make terminology slightly confusing. Indeed, 1 is a vector (element) in the vector space of all quaternions, but is not a "vector quaternion", the ''vector quaternions'' being those quaternions for which <math>a=0;</math> they form a vector space of dimension 3. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:43, 15 February 2024 (UTC)
:* The [[metric]] on SU(2) aka S_3 aka CP(2) is called the ... crap, I don't remember the name. Oh right [[Fubini-Study metric]]. Its basically just the standard metrix on the sphere, but it looks interesting when you write it out for the typcal notation used in QM and in [[algebraic geometry]] which each have thier unique notiations (alg. geom studies complex projective spaces).
 
== What the eff? ==
:* One of the things that confuses people is the relationship between SU(2) and su(2) because both use 2x2 complex matrices. They're not the same tho, because SU(2) is a group, su(2) is an algebra. Likewise, there is a similar confusion for quaternions: There is the algebra H and there is the group H and they both use 1, i, j, k to understand how these differ, it helps to keep su(2) vs SU(2) firmly rooted in mind.
 
The section on "P.R. Girard's 1984 essay..." is full of references to the author. I'm too busy, but someone needs to clean that up or delete the entire ugly self-promotion. [[user:Verdana_Bold|Verdana<span style="color:red;">♥</span>'''Bold''']] 10:52, 25 February 2024 (UTC)
:* Wait -- there's more... if you allow the vector <math>\vec{v}</math> to be complex, then you get representations of the group [[SL(2,C)]] which have [[SO(3,1)]] as a representation -- this is the group of [[special relativity]]. which is why the [[outer product]] of two relativistic spinors is a spin-1 [[boson]]. Add the sqrt(-1) and you can say the same with quaternions, if you wanted to. You could write out [[Einstein's equation]]s for [[general relativity]] using quaternions, if you wanted to. This is because the quaternions are <math>sqrt{-1}</math> times the usual generators of sl(2,C). People have actually done this: its vaguely instructive to see those eqns as SL(2,C) instead of text-book standard SO(3,1).
 
:Paragraph removed. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:07, 25 February 2024 (UTC)
:* The [[outer product]] of two quaternions gives the [[Runge-Lenz]] vector: it describes the orbital mechanics of planetary systems (planets orbiting a sun) and the conserved qauantities are given by [[SO(4)]] (not just SO(3)).
::Wait, why delete this section. From what i see and find, the references are indeed correct. How can this be self-promotion? [[User:Mwcb|Mwcb]] ([[User talk:Mwcb|talk]]) 12:01, 2 March 2024 (UTC)
:::Details on a specific author does not belong to this article. For not being promotional, an independent source is needed. Such a source must discuss the importance, if any, of the results. Without that, the paragraph is there only for promoting an author. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:38, 2 March 2024 (UTC)
 
== Extend ==
:* The restriction of SL(2,C) to real numbers gives [[SL(2,R)]] which is a [[hyperbolic manifold]] important in [[number theory]] and seems to have somethhing to do with the [[Riemann hypothesis]]. This is one of the many [[Siren (mythology)|Siren's call]]s that [[string theory|string theorists]] are unable to resist. (well, string [[world sheet]]s are [[Riemann surface]]s which leads to [[conformal field theory]] and [[AdS/CFT]] correspondance which gives [[monstrous moonshine]]. Heh. So there.
 
What does the term "extends" mean in the first sentence of this article? [[User:Comfr|Comfr]] ([[User talk:Comfr|talk]]) 18:52, 17 April 2024 (UTC)
:*In short, when you really start fucking with it, you find all of these low-dimensional concepts are isomorphic or homomorphic to each other, which makes for a very rich playground of things related to each other.
 
:It means that the complex numbers can be considered as a subset of the quaternions, with the same behavior as quaternions that they have as complex numbers, but quaternions also include additional elements which combine compatibly with the existing ones. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 20:33, 17 April 2024 (UTC)
:Anyway, I clearly had too much fun writing the above. Thanks for posing the question. [[Special:Contributions/67.198.37.16|67.198.37.16]] ([[User talk:67.198.37.16|talk]]) 01:39, 13 February 2015 (UTC)
::Because this is a technical article, technical terms should be well defined. Unfortunately, the technical term "extends" appears in the article without a definition, which motivated me to hyperlink "extends" to [[Field extension]].
::[[User:Quantling]] correctly observed that Quaternion are not a field. Quaternion are not commutative, which is a required property of a field.
::Vector products and also not commutative, however "vector fields" exist.
::The article [[Field extension]] does not say that a field extension might not be a field. Should that be fixed? [[User:Comfr|Comfr]] ([[User talk:Comfr|talk]]) 22:14, 20 April 2024 (UTC)
:::See [[ring extension]] for an appropriate link.— [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 00:01, 21 April 2024 (UTC)
:::Two more on topic: [[Complexification#Dickson doubling]] and [[Cayley%E2%80%93Dickson construction]]. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 00:09, 21 April 2024 (UTC)
:::To clarify: there are (at least) two meanings of ''field'' in mathematics. In ''algebra'' it means a set with a commutative addition and commutative multiplication operation and various additional properties; this is the present discussion. In ''analysis'', it means a function defined on a manifold (including manifolds like ordinary Euclidean spaces); and if the result (image) of the function is a vector (at each point of the manifold) then it is called a ''vector field''. —[[User:Quantling|<span class="texhtml"><i>Q</i></span>uantling]]&nbsp;([[User talk:Quantling|talk]]&nbsp;&#124;&nbsp;[[Special:Contributions/Quantling|contribs]]) 13:52, 22 April 2024 (UTC)
:::In this context, "extends" does not need to be interpreted as a precise technical term; the ordinary English meaning of the word is plenty clear. I would not bother wiki-linking it to anything. –[[user:jacobolus|jacobolus]] [[User_talk:jacobolus|(t)]] 18:24, 22 April 2024 (UTC)
 
An underlying extension is the [[group extension]] from {1, i, –1, –i } ≅ ℤ<sub>4</sub> to the [[quaternion group]] ℚ<sub>8</sub>. The extension is not uniquely determined and can lead to the [[dihedral group of order 8]] which lies under the [[coquaternion]] ring. [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 00:09, 28 April 2024 (UTC)
::I see the question served its purpose: Ozob addressed it with [https://en.wikipedia.org/w/index.php?title=Quaternion&diff=prev&oldid=617704349 this edit]. Yes, the connections are varied and deep, and fun if you live with this stuff. —[[User_talk:Quondum|Quondum]] 03:52, 13 February 2015 (UTC)
 
Overheard in a 19th century classroom:
== Terminology: "scalar part" and "vector part" ==
:'''Student''': We are familiar with complex numbers, but now you want to introduce some things even more complicated than these complex ones. What do you propose to call such things?
:'''Teacher''': Well, [[hypercomplex number]]s, of course. — [[User:Rgdboer|Rgdboer]] ([[User talk:Rgdboer|talk]]) 01:02, 5 May 2024 (UTC)
 
== Errors in product graph image ==
I find this terminology inept and ugly. Why not stick with real and imaginary parts?
 
The image showing the cycles of multiplication appears to be incorrect. In particular, the arrows in the three outer cycles should be inverted.
Is that terminology standard in the literature (the maths literature, not the physics one)? Not if I trust the few papers I've read today, but I could be wrong.
If it is standard, then I guess some could argue that Wikipedia should continue to spread this ugliness. Otherwise it would be a shame that Wikipedia helps set up or propagate an ill-suited standard for no reason.--[[User:Seub|Seub]] ([[User talk:Seub|talk]]) 05:59, 11 March 2015 (UTC)
 
For example, starting at positive j, cycling along the blue path counter clockwise (xk):
: Real and imaginary parts is used in complex numbers where the parts are equal in dimension. scalar and vector parts emphasises that they are not equally sized, but a 1-dimensional scalar part and a 3-dimensional vector part. And they are vectors in a very real sense too; vector operations such as the cross product, dot product scalar product arise from quaternion product by considering how the 'vectors' in them multiply.--<small>[[User:JohnBlackburne|JohnBlackburne]]</small><sup>[[User_talk:JohnBlackburne|words]]</sup><sub style="margin-left:-2.0ex;">[[Special:Contributions/JohnBlackburne|deeds]]</sub> 12:41, 11 March 2015 (UTC)
 
* j * k = -i (graph shows positive i)
::I think that there are arguments both ways. Doesn't the ''scalar''/''vector'' terminology date to Hamilton, whereas ''real''/''imaginary'' and other variants tend to be more contemporary? I am not particularly a fan of the use of "vector part", because it does not seem to generalize directly to [[hypercomplex number]]s or [[Clifford algebra]]s, and would be destined to fade. John is correct that it does relate directly to vector algebra, but only in exactly 3 dimensions. IMO, it would be more appropriate to speak of the "scalar" and "nonscalar" parts. At least "real" can "imaginary" do fit as a generalization of the use of the concepts from complex numbers, and I've seen the term "imaginary" used to refer to any nonreal component that squares to a real number, such as in [[split-complex number]]s. For the purposes of this article, perhaps we could switch to "real" and "nonreal" (just a suggestion)? —[[User_talk:Quondum|Quondum]] 17:54, 11 March 2015 (UTC)
* i * k = j (graph shows negative j)
* -j * k = i (graph shows negative i)
* -i * k = -j (graph shows positive j)
 
The same is true for the outer red and green cycles. However, inverting the direction fixes the error. [[Special:Contributions/2620:1F7:93F:425:0:0:32:14F|2620:1F7:93F:425:0:0:32:14F]] ([[User talk:2620:1F7:93F:425:0:0:32:14F|talk]]) 12:51, 6 June 2024 (UTC)
:::At the level of this article vector algebra is done in 3D. E.g. you have the cross product, defined only in three dimensions, as a key operation. This and the dot product show how it is a 'vector' part as both operations arise directly from the quaternion product if restricted to products only of the vector parts. At least that's how I learned it. I later learned how to generalise it in various ways, which leads to other ways to think of the non-scalar parts, as imaginaries, as bivectors . But vectors I think is most usual at a less advanced level.--<small>[[User:JohnBlackburne|JohnBlackburne]]</small><sup>[[User_talk:JohnBlackburne|words]]</sup><sub style="margin-left:-2.0ex;">[[Special:Contributions/JohnBlackburne|deeds]]</sub> 00:21, 12 March 2015 (UTC)
 
:The diagram is correct. As you can see in several places in the article (for example, [[Quaternion#Multiplication of basis elements|Multiplication of basis elements]]), '''jk''' = '''i''', '''ik''' = –'''j''', –'''jk''' = –'''i''', and –'''ik''' = '''j'''.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 13:25, 6 June 2024 (UTC)
::: Just because quaternions can be related to vectors and rotations in Euclidean 3-space doesn't mean that it is their nature or purpose, it's just a use you can make of them. So I disagree with the argument that the imaginary part of a quaternion is a vector in a "very real sense". And the "scalar part" term is no more clever imho, why would you want to think of it as a scalar? It is just a quaternion that is identified as a real. I recall that scalars are what you name the elements of the base field of a vector space, when you think of them acting by multiplication. I also recall that in (really not so) modern mathematics, vectors don't refer to arrows in the plane and the 3-space that you can dot product and cross-product. My recommendation is "real" and "imaginary", I see no drawback in that terminology. I also think it's the more common usage in modern mathematics (we could survey the arXiv a little), but I'm not certain of that.--[[User:Seub|Seub]] ([[User talk:Seub|talk]]) 10:17, 12 March 2015 (UTC)
Add topic
Retrieved from "https://en.wikipedia.org/wiki/Talk:Quaternion"