Talk:Quaternion

From formulasearchengine
Jump to navigation Jump to search

Template:Maths rating User:MiszaBot/config Template:OnThisDay Template:Archive box

regarding basis showm in Matrix representations

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? Cplusplusboy (talk) 13:22, 20 January 2012 (UTC)

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. Rschwieb (talk) 13:58, 20 January 2012 (UTC)
Arbitrary 4 × 4 real matrix without Jordan blocks with same eigenvalues (namely, {Template:Mvar, Template:Mvar, −Template:Mvar, −Template:Mvar} ) is eligible to represent the quaternion Template:Mvar. You may construct real 4-dimensional quaternions' representations by algebraic conjugation: Template:MvarTemplate:Mvar−1Template:MvarTemplate:Mvar where Template:Mvar is a canonical representation and Template:Mvar 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. Incnis Mrsi (talk) 14:30, 20 January 2012 (UTC)
(Note to OP: the conjugation described here produces an automorphism of H. Rschwieb (talk) 15:12, 20 January 2012 (UTC)
It is an automorphism of ℍ only if Template:Mvar 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, Template:Mvar does not even have to be orthogonal, this means that Template:Mvar−1Template:MvarTemplate:Mvar not necessary is a canonical representation of any quaternion. Incnis Mrsi (talk) 16:20, 20 January 2012 (UTC)
Oh. I've never heard of a reversible matrix, so I was guessing it meant special orthogonal. Rschwieb (talk) 20:25, 20 January 2012 (UTC)
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 S2? — Quondum 05:18, 21 January 2012 (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. — 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. 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. — Quondum 06:35, 22 January 2012 (UTC)

Difference between quaternions and vectors.

To me it is not quite clear what the difference is between quaternions and 3 dimensional vectors. It seems as if you could define any n dimensional number system, you could say a number of the form ai+bj+ck+dl+em could describe certain properties of a 5 dimensional space. Do quaternions really 'exist' or are they definitions of an alternative way to describe things you could describe with vectors? Complex numbers have a meaning in the sense that they can be square roots of negative numbers, you can solve certain equations with them that you couldn't solve with real numbers.

Do quaternions have comparable arithmetic, or are they just an alternative way of describing 3 dimensional vectors? Maybe it is good of this article addresses this and explains the differences and the exact meaning of quaternions. — Preceding unsigned comment added by 83.160.61.76 (talk) 13:19, 12 April 2012 (UTC)

Quaternions are four dimensional vectors (technically, they form a vector space of dimension 4 over the real numbers), but they are more than this because they also come equipped with operations of multiplication and division. This means they are not just vectors, they are a division algebra. Gandalf61 (talk) 13:29, 12 April 2012 (UTC)
There isn't really a "difference", nor are they the same. Vectors are elements of a vector space, and quaternions happen to be a vector space. They are also more than a vector space because they form a division ring. Most vector spaces don't carry an associative multiplication, as the quaternions do.
You could say the same for the real n-by-n matrix algebra. It is an n2 dimensional vector space, each matix itself a vector. There isn't a "difference" between matrices and vectors, but the algebra does have a particular multiplication (matrix multiplication) specified. Rschwieb (talk) 16:37, 12 April 2012 (UTC)
You both seem to have missed to OP's point about representing 3D geometric vectors. Quaternions are a pretty natural algebra for 3D vectors, not 4D (even though pairs can represent a rotation in 4D). Though being a very "real" algebra in its own right, quaternions can be used for 3D vectors much like in "standard" vector calculus, except that rotations are more natural. — Quondum 16:59, 12 April 2012 (UTC)
Wow, I can't wait to spring that back on you when you attempt to answer a vague post.smile I have to admit, I cannot understand the original question. It looks like "what is the difference between apples and fruits?" Help the 2/3 of point-missing posters understand what interpretation makes them confusingly similar to vectors. Rschwieb (talk) 00:07, 13 April 2012 (UTC)
Okay, feel free, I guess I deserve it. blushing If you consider the section Quaternion#Quaternions and the geometry of R3, you'll see some motivation for what I say. Historically, Hamilton used quarternions for and and was motivated by manipulation of 3D geometric vectors; it was a competing (and out-competed) alternative to Gibbs's vector operations. This is behind the naming of the "vector part" of a quaternion. Reading between the lines, it seems to me that the OP intended this interpretation, and if I'm correct, the response highlighting only the 4D vector space quality is almost certain to confuse the asker. — Quondum 12:45, 17 April 2012 (UTC)
I'm positive you interpreted it correctly because I trust your familiarity with these this topic. I suppose I have to blame myself for not finding time to read this article all the way through, yet. I'll direct future questions on the topic to your talkpage.
I did just see something that excites me though:

He concluded by stating "I firmly believe that quaternions can supply a shortcut for pure mathematicians who wish to familiarize themselves with certain aspects of theoretical physics."

^^ exactly what I'm looking for! Rschwieb (talk) 13:14, 17 April 2012 (UTC)

Using the vector part of a quaternion to hold a 3D vector doesn't give you anything much. Where they really shine is in the way they can handle a rotation about a point and that's why they are used in 3D game engines for instance. That's what the last sentence in the lead is referring to though the article, see Quaternions and spatial rotation about this. Dmcq (talk) 16:52, 17 April 2012 (UTC)

Pronunciation

I know this is a trivial point, but how is the word 'quaternion' pronounced? In English the vowel 'a' following 'qu' can have at least six pronunciations: short a as in 'quack'; the 'ah' sound as in the traditional British pronunciation of 'qualm'; the 'o' sound, as in 'quaff' or 'quash'; the 'aw' sound as in 'quarter'; the long 'a' sound as in 'quake'; and the indeterminate vowel sound as in 'equatorial'. (I'm aware that some or all of these may vary as between British and American pronunciation and indeed between regions within Britain and America.) Do mathematicians have a uniform pronunciation of the word, and if so what?86.183.202.149 (talk) 16:21, 11 March 2013 (UTC)

Since it is clearly using the same root as "quarter" is, I would imagine most English speakers prounounce the "a" in the same way. That is the case for all the US speakers that I know of. Rschwieb (talk) 17:10, 11 March 2013 (UTC)
Incidentally, I am not aware of any sound difference between "qualm" "quaff" and "quash". They all sound like the same vowel to me. Rschwieb (talk) 17:12, 11 March 2013 (UTC)
"qualm" rhymes with "calm"; "quaff" rhymes with "cough" Gandalf61 (talk) 17:36, 11 March 2013 (UTC)
Maybe this differs as you cross the Atlantic, but I've always assumed "quaternion" had the short 'o' sounds as in "quaff"/"cough"; fairly sure I've heard it pronounced that way in the UK. Gandalf61 (talk) 17:36, 11 March 2013 (UTC)
According to the OED:
  • Quaternion: Brit. /kwəˈtəːnɪən/ , U.S. /kwəˈtərniən/ , /kwɑˈtɛrniən/
  • Quaff: Brit. /kwɒf/ , U.S. /kwɑf/
  • Qualm: Brit. /kwɑːm/ , U.S. /kwɑ(l)m/ , /kwɔ(l)m/
Presumably, see International Phonetic Alphabet and International Phonetic Alphabet chart for English dialects to find out how to turn these glyphs into actual sounds. Jheald (talk) 17:58, 11 March 2013 (UTC)
(Incidentally, per the OED entry quaternion is first recorded in Wycliffe's Bible of 1384, where Peter in the Acts of the Apostles is put into the hands of four quaternions of soldiers -- ie four squads of four men each). Jheald (talk) 17:58, 11 March 2013 (UTC)
@Gandalf61 OK, maybe it is the difference between "all" and "off" :) Without that example, I would have a hard time hearing they are different.
@Jheald I think what you've listed has identified a second pronunciation that sounds familiar to me: "kwuh". With "kwah", these seem like the most familiar US pronunciations. I think regional dialect might also cause "kwatt". But never, as far as I can tell, "kway". Rschwieb (talk) 18:21, 11 March 2013 (UTC)

-1 in the multiplication table

In the multiplication, should -1 be included? I guess it is sort of self explanatory, but 1 is even more simple. TheKing44 (talk) 18:29, 2 August 2013 (UTC)

Error?

I've never made an edit (except for the occasional spelling fix) so not sure of protocol.

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)

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 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 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. 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 Rn) 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. Ds1392 (talk) 01:19, 21 October 2013 (UTC)
I don't think it can be clarified without properly distinguishing what "dimensions" are being discussed. It has (geometric) dimension 4 when embedded in R4, but has (topological) dimension 3. 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. 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. 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. 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. Ozob (talk) 18:44, 21 October 2013 (UTC)
I've changed it to "3-sphere S3". 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. — 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-R3 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 :) Ds1392 (talk) 12:13, 23 October 2013 (UTC)