Talk:Multivector

Merge with p-vector

As the two things are different, it would be preferable to have two articles that link to each other. Brews ohare (talk) 20:35, 4 January 2010 (UTC)

Hi Brews,

Elie Cartan, who coined differential forms, uses "Multivector" and "p-vector" as synonyms on "p-vector" p. 16 of "The theory of spinors", calling the section "Multivectors" and referring to a multivector of a given degree p as a p-vector. Thus I've merged the articles under "Multivector", following Cartan: a multivector is an element of the exterior algebra on a vector space, while a p-vector is an element of degree p.

What distinction do you see between the concepts?

—Nils von Barth (nbarth) (talk) 05:31, 13 February 2010 (UTC)

Ok, I think the distinction you're drawing is between:

• Simple/decomposable elements/blades (a^b^c)
• Homogeneous elements (single degree: a^b + c^d)
• Elements of mixed degree (a + b^c)

This is a useful and confusing distinction, and all senses seem to be used (terminology seems pretty inconsistent). If there's much that can be said about each of these specifically, I'd be happy to have separate articles (clearly distinguishing), otherwise (and until that time) I'd prefer to have a single article, so it's not too scattered. How does this sound?

—Nils von Barth (nbarth) (talk) 06:08, 13 February 2010 (UTC)

As per discussion here, the above seems to be correct - multivector is used (narrowly) to mean "elements of mixed degree", but the terminology is inconsistent, and, to the extent that these are just terminology, this article works ok as a summary (of names and important examples), leaving details to specialized articles (bivectors, pseudovectors, pseudoscalars, etc.).

—Nils von Barth (nbarth) (talk) 19:23, 13 February 2010 (UTC)

Blade

I've just changed "A sum of only n-grade components is called an n-vector, or a homogeneous k-blade." to "A sum of only n-grade components is called an n-vector, or a homogeneous multivector." All other sources I've seen, incliding the reference cited here, describe blades are *simple* multivectors, i.e. a scalar or a wedge product of some number of the generating vectors of the algebra, rather than a sum of wedge products. Dependent Variable (talk) 11:43, 14 September 2010 (UTC)

Changes not helping

The lede has become confusing. The statement "A multivector that is a linear combination of basis multivectors constructed from p basis elements is called a rank p multivector, or a p-vector" is difficult to interpret in the way it is intended (what is "constructed from"?). "A vector space with a vector multiplication is called an algebra" is unnecessarily imprecise. The term "vector space" is used in the lede to describe different subspaces of the algebra without highlighting the distinction. Use of the term "rank" is inconsistent with elsewhere in WP, most relevantly with Wedge product#Rank of a k-vector); we use the term "grade" in this context. Including illustrations of the wedge product in the body is unnecessary duplication; I would not like to see WP become an endless series of repetitive tutorials: Wikipedia is not a textbook. Although the article was and still is sorely incomplete, the recent changes have not, IMO, been an improvement. —Quondum 17:24, 26 March 2014 (UTC)

For what it's worth here is the comparison: [1]. M∧Ŝc²ħεИ_τlk 17:33, 26 March 2014 (UTC)

Perhaps part of the problem is that the term is very context-specific – i.e., it is not a standalone concept. The lead should first establish the context, which is exterior algebra or geometric algebra. The basic definition is as a general element of the algebra (i.e. not as built up in any independent way). The article should not try to define these algebras, but it should highlight that multivectors are linear combination of k-vectors of different k, which in turn are linear sums of k-blades (i.e. k-vectors of rank 1). Sometimes the term k-multivector is used in place of k-vector. I think that the intention of making it more readable is achievable, but it would be a pity to lose understandability due to the wording becoming too fuzzy in the process. —Quondum 18:14, 26 March 2014 (UTC)

I welcome all suggestions. However, please notice that exterior algebra generally refers to differential forms and geometric algebra is a non-traditional formulation of Clifford algebra. These are distinct from what is a relatively simple concept of a multivector, which is the product of two vectors. The use of the word rank for multivectors is traditional, where as the term grade arises from its structure as a graded algebra. I purposely have left the original lead in place, so please cut and paste so it fits your preferences. By the way, a Clifford number is a distinct object, so introducing that concept at this stage is not helping. Prof McCarthy (talk) 18:42, 26 March 2014 (UTC)

I might add a couple of points. There is only one vector space that provides support for the construction of multivectors. The resulting multivectors form additional vector spaces. And if this is not clear, then some simple edits should take care of that. I might add that a vector space with a linear product operator for vectors is called an algebra, which is where the terms matrix algebra and vector algebra come from. I am not sure how this can be imprecise. Prof McCarthy (talk) 18:50, 26 March 2014 (UTC)

By illustrations, I assume you mean the descriptions of multivectors in R2 and in R3. The important feature of multivectors illustrated in these two sections is their direct connection to the measurement of area and volume in all dimensions. This is what these examples demonstrate. So I disagree with your opinion on this point. But, because I am making the revisions is it the case that your opinion must stand? I have had trouble with this aspect of Wikipedia in the past.Prof McCarthy (talk) 18:58, 26 March 2014 (UTC)

I have tried to address the concerns with a number of revisions. Please be patient with the repetition of the bivector an three vector calculations from the section on exterior algebra. My goal is to introduce multivectors on projective spaces to motivate their importance as coordinates for hyperplanes, which is how they were originally formulated before being adapted to calculus through reformulation as differential forms. Prof McCarthy (talk) 21:20, 26 March 2014 (UTC)

My opinion stands only as an opinion. I apologize for my tone. We haven't had opinions from others, and I'm relatively uncertain on several aspects, so it's not going to help for me to interfere. —Quondum 05:13, 27 March 2014 (UTC)

Thank you. I will be attentive to your concerns. Prof McCarthy (talk) 13:16, 27 March 2014 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Multivector/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Geometry guy 22:34, 21 May 2007 (UTC) This comment applies to "Multivector" and to "P-vector" which both link here. I suggest those parts of both pages that specifically relate to Geometric Algebra could usefully be added to that page and deleted from these pages. Terms like 2-vector, multivector and grade (of a (k)-vector) suffer from confusing use across different fields. 213.192.200.2 (talk) 15:07, 23 March 2009 (UTC)

Last edited at 15:07, 23 March 2009 (UTC). Substituted at 02:21, 5 May 2016 (UTC)

Trivector

So in 3 dimensions every trivector can be represented as a scalar times the unit trivector e₁ ∧ e₂ ∧ e₃. So what is the rank 3 tensor form of e₁ ∧ e₂ ∧ e₃? Most of the entries will be zero but which ones will be one and which ones will be negative one?

$e_{1}\wedge e_{2}\wedge e_{3}={\begin{bmatrix}0&0&0\\0&0&1\\0&-1&0\\\end{bmatrix}}{\begin{bmatrix}0&0&-1\\0&0&0\\1&0&0\\\end{bmatrix}}{\begin{bmatrix}0&1&0\\-1&0&0\\0&0&0\\\end{bmatrix}}$

Just granpa (talk) 08:48, 14 November 2017 (UTC)

A tensor with elements of the Levi-Civita symbol ($\varepsilon _{ijk}$) might be what you are looking for. So it’s 1 for even permutations of i, j and k, -1 for odd permutations, and zero if any two of i, j and k are the same.

But that does not so much represent the unit trivector as the product itself. If you then take its product with any three vectors a, b, c you get the trivector a ∧ b ∧ c. So to get e₁ ∧ e₂ ∧ e₃ take its product of this tensor with e₁, e₂ and e₃.--JohnBlackburne^words_deeds 12:56, 15 November 2017 (UTC)

So as I suspected there is a nonzero entry for every permutation of 1, 2 and 3.

123, 132, 213, 231, 312, 321.

Just granpa (talk) 14:11, 15 November 2017 (UTC)

Yes, something like that. But it represents the product, not the result. The result e₁ ∧ e₂ ∧ e₃ is the unit pseudoscalar, a scalar for most purposes. I think that makes it a rank zero tensor, rather than rank 3.--JohnBlackburne^words_deeds 14:22, 15 November 2017 (UTC)

In 3 dimensions its dual is a scalar.

$a\wedge b\wedge c=a\otimes b\otimes c-a\otimes c\otimes b+c\otimes a\otimes b-c\otimes b\otimes a+b\otimes c\otimes a-b\otimes a\otimes c$

Just granpa (talk) 14:43, 15 November 2017 (UTC)

Just granpa, I just found something that may be of interest: Tensor Representation of Geometric Algebra. It’s similar to my thinking, except instead of just taking the product of three vectors, they take the product of all elements of a GA, which becomes rather unwieldy. E.g. the example, the simplest non-trivial GA G₂, needs a 4 x 4 x 4 tensor. For three dimensions you would need a 8 x 8 x 8 tensor. And so on. Not especially practical or useful, but another way of doing it I guess.--JohnBlackburne^words_deeds 01:54, 22 November 2017 (UTC)

Merger proposal

I propose to merge Blade (geometry) into Multivector. I think that the content in the Blade (geometry) article can easily be explained in the context of Multivector, and the Multivector article is of a reasonable size that the merging of Blade (geometry) will not cause any problems as far as article size is concerned. Rgdboer (talk) 00:47, 10 January 2019 (UTC)

This topic is part of multilinear algebra, an extension of linear algebra. The term blade is used to mean p-vector by David Hestenes in one of his books. According to his article, Hestenes acknowledges Grassmann as father of multilinear algebra. A separate article for Blade (geometry), when it is a multivector, is superfluous and deceptive to students looking for proper terminology. Project support for pejorative and unnecessary terms undermines the reliability of the encyclopedia. — Rgdboer (talk) 01:32, 11 January 2019 (UTC)

The following was removed:

The term geometric algebra was used by E. Artin for matrix methods in projective geometry.<ref E. Artin, Geometric algebra, Interscience Publ., 1957 /ref>

The book is Geometric Algebra and has nothing to do with multivectors. Sweetening with mention of Artin is deceptive. — Rgdboer (talk) 02:46, 13 January 2019 (UTC)

Review of the article flag (geometry) shows that in a graded structure there may be recourse to elements (multivectors) with terms of different grades. Since a blade excludes such a multivector, it appears that the terms blade and multivector are distinguishable. The Merger suggestion is being removed. — Rgdboer (talk) 01:31, 5 May 2019 (UTC)