Talk:Sesquilinear form

page split?
I'm considering splitting this page into sesquilinear form and hermitian form, which currently redirects to the former. Does anybody object? Markus Schmaus 15:22, 27 July 2005 (UTC)


 * I certainly don't. As a matter of fact, this article shouldn't immediately zoom in on the complex case either, one can easily consider sesquilinear forms over finite fields Evilbu 01:38, 12 February 2006 (UTC)


 * I never got round to actually do this.
 * I've never seen sesquilinear forms over finite fields, I only know the complex case. But go ahead and change the article the way you think. Markus Schmaus 18:00, 12 February 2006 (UTC)


 * I'd rather keep the hermitian form stuff on this page (unless the material grows too large). We discuss symmetric bilinear forms on the bilinear form page. Sesquilinear forms over fields other than C should be mentioned, but the complex case should be emphasized, as it is by far the most important. -- Fropuff 18:12, 12 February 2006 (UTC)


 * Let me add some elements here. First of all I took the liberty to stop the symmetric bilinear form page from redirecting simply to the bilinear form page.  My main reason is that things like orthogonal basis, real and complex cases and sylvester's inertia law, and the connection with orthogonal polarities cannot all be explained in the bilinear form section (they really are to me a class apart).
 * I understand why you wan't to make it so clear that the complex case is so important, but for finite geometers the finite case is just as important, and making a new page for that is highly confusing. I must say I haven't seen that many relevant sesquilinear forms over finite fields that aren't hermitian.
 * Last : I think hermitian forms definitely need to be separated. Unitals, unitary polarities and varieties, you can't do that all on the sesquilinear forms page.Evilbu 22:35, 14 February 2006 (UTC)
 * If you are going to add this content, this is definetly enough to justify seperate pages. The finite case is definitly interesting, I only wish, that the complex case comes first. If an undergrad student bumps into this page and sees at the first glance a definition involving finite fields, he's probably confused. A person interested in the finite case probably already heard about the complex case and just scrolls down. Markus Schmaus 00:49, 15 February 2006 (UTC)

Standard Hemitian form
I appreciate that the article does mention that physicists and mathematicians conventions differ at the top, and states which it follows, but if one is looking for the definition of a Hermitian form, as I was, and skipped the first section, as I did, then those following the more common mathematical convention are going to be surprised or confused. How about, at least for the standard Hermitian form definition adding, "or for those following the opposite convention to this article," and then the standard form for that convention. I know it should be obvious, but I think it would be helpful and would help alert any reader who has skipped straight to that section, that they've probably missed something important.--HugoBarnaby (talk) 23:05, 11 August 2009 (UTC)

The first argument should be linear
The definition of sesquilinear is certainly that the first argument is linear. The physics notation is confusing and backwards and is really only ever used for the hermitian form anyway. Also, the * operator is typically the one used for the dual space, and the bar is usually used for a closure operator or to denote residue classes. This article should be fixed. 67.194.132.91 (talk) 20:32, 15 September 2009 (UTC)


 * I think that which argument is linear is still open for debate. The noncommutative case is significant, and an article such as this should perhaps adopt a convention that uniformly treats both commutative and non-commutative cases, while clearly calling out existing conventions.  In the noncommutative case, the conventions that I have seen always bring the scalar multiplier (with or without the anti-automorphism) out on the adjacent side: φ(αu,βv)=αφ(u,v)σ(β) or φ(uα,vβ)=σ(α)φ(u,v)β (but don't take my word for it – I have not read widely).  Which has the anti-automorphism applied is then inherently determined by whether the module is a left or right module.  There seems to be a mix of preferences for left or right modules in mathematics.  Column vectors and kets are naturally right modules; in this context, even to a mathematician the physicist's convention makes sense.  If my argument here is accepted, then I think it comes down to a choice between preferences of presenting this in terms of left or right modules, and (assuming the adjacency convention above) not about which argument.  I generally like to use right modules, since these correspond to column vectors and matrix multiplication from the left; one just has to get used to the idea of scalar multiplication from the right.  But if someone feels that left modules and row vectors (and consequently having a multiplying matrix on the right) is the norm for presenting this topic, please discuss this here.   —Quondum 18:55, 1 September 2016 (UTC)

Equivariance
Here's something that confuses me about the article. It may just be that I'm new to these concepts and so don't understand something the author took for granted.

This article states:


 * Bilinear forms are to squaring (z^2), what sesquilinear forms are to Euclidean norm ( |z|^2 = z * z).


 * The norm associated to a sesquilinear form is invariant under multiplication by the complex circle (complex numbers of unit norm), while the norm associated to a bilinear form is equivariant (with respect to squaring).

The Equivariance article states:


 * In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. Specifically, let G be a group and let X and Y be two associated G-sets. A function f : X --> Y is said to be equivariant if


 * f(g.x) = g.f(x)

Guessing that the dot in the above equation means the group operation, and that G-sets means subsets of the underlying set of G, and that "for all g in X and x in Y" is implicit in this (special re)definition of "commutes", wouldn't all of this lead to the following meaningless equality? Let f be the quadratic form that gives what the first of these articles call the Euclidean norm, namely the square of what is conventionally called the Eucludean norm. Then if f is equivariant in any way according to the article Equivariance (regardless of what this article might mean by "with respect to squaring"),

f(g+x)=g+f(x) ??

In other words, using the conventional dot-product notation, it seems to be claiming that for all vectors g and x,

(g+x).(g+x)=g+x.x ??

Obviously this isn't the intended meaning. Could it be worded more clearly, or is it only my unfamiliarity with the concepts that's to blame? Perhaps the article doesn't mean to suggest that its "Euclidean norm" is the square of the more conventional Euclidean norm. Perhaps it means a different and more general kind of equivariance from that defined in the article of that name, or perhaps I was wrong to assume the group in question is the vector space. Dependent Variable (talk) 13:41, 11 June 2010 (UTC)

"Standard" for mathematicians
I added a [citation needed] tag to the statement about the standard for mathematicians. I'm getting my PhD in math, and the first and only time I've ever seen linearity in the first term and conjugate-linearity in the second term is in various pages here on Wikipedia. --Yoda of Borg (talk) 08:46, 5 November 2013 (UTC)


 * I don't have a lot of references handy, but Rudin's standard book Real and Complex Analysis (that's "green Rudin") and Stein and Shakarchi's newer textbook Fourier Analysis both take the inner product to be linear in the first term, not the second. This should be visible anywhere that a book defines $$L_2$$. &mdash; Carl (CBM · talk) 13:24, 5 November 2013 (UTC)

Too many hyperlinks!!!
I found it very hard to engage with this article. Just too much assumed knowledge. Linear, antilinear, bilinear, linear in one argument, antilinear in the other. This is barely English intended to convey meaning. I have a quite reasonable mathematical background, yet collapsed under the weight of jargon. I suffered death by hyperlink, which is to say that by the time one has looked up all the hyperlinks, and chased down their hyperlinks, one is exhausted and confused.

As a teacher I would write 'Can do better!' on the report card.

Instead, can we provide a diagram that shows the nature of the mapping, and compares linear with antilinear, on the page, rather than associated with hyperlinks?

The article DOES tell us that sesqui means "one and a half". And...? Why did they tell us that? One and a half linear. Helpful.

Centroyd (talk) 00:29, 16 February 2014 (UTC)centroyd


 * Yes. I bet you are here because you clicked on Hermitian form. This article should start by mentioning the standard Hermitian form, and then generalizing and abstracting. 67.198.37.16 (talk) 20:06, 4 December 2020 (UTC)

Generalization to modules

 * "Sesquilinear forms are not restricted to the complex numbers and may be defined on any K-module where K is a division ring"

Though I'm not disputing the correctness of the statement, I'm wondering about its naturalness. In particular, a sesquilinear form is presumably determined by an anti-automorphism, and unless there is a "natural best choice" of anti-automorphism on every division ring, this statement should be qualified by the mention of "given an anti-automorphism". The choice of anti-automorphism in the case of complex numbers is an interesting case (presumably, every anti-automorphism is also an automorphism in a commutative ring, and there are apparently an infinite number of these in C), where the choice is externally imposed via the "canonical" embedding of the reals in the complex numbers).

Further, why is there a restriction on this statement to division rings? Doesn't it apply to all rings? —Quondum 23:32, 25 April 2015 (UTC)


 * This is coming from the projective geometry literature. I was trying to avoid being too technical in the lead and just wanted to make the point that even though the article is written in terms of the complexes, the concept is more general than that. Although I haven't checked it out myself, I've never seen these forms defined over anything more general than division rings (and usually limited to fields), so I suspect that zero divisors will get you into trouble. The article's point of view is a bit slanted as none of its references single out the complexes (also note the lack of in-line citations until I added some today). Baer's work is fundamental but you now need a glossary in order to read him. I'd like to expand this section with some more modern citations and amplify the natural connection with dualities of projective spaces. Bill Cherowitzo (talk) 03:19, 26 April 2015 (UTC)


 * Bourbaki talks about
 * "The mapping (x, x*) ↦ (x, x*) of E × E* into A is called the canonical bilinear form on E × E* (the notion of bilinear form will be defined generally in IX, § 1)." (Algebra II, §2.3)
 * There is no restriction to commutativity of the base ring A nor to skew fields, nor on the type of A-module E. The promised chapter (IX), according to the index is about "Sesquilinear and quadratic forms". So general rings seem to be suitable; I read nothing into the restriction to a module and its dual in this context (that is what makes the bilinear form canonical). The unfortunate part about mentioning restrictions (e.g. to skew fields or commutative rings) is that they may be thought of by a reader as a necessary restriction rather than as a deficiency of sourcing. My own guess is that for the case of complex numbers and others, sesquilinear forms are just an ugly replacement for using a bilinear form over a vector space and its dual – exactly what Bourbaki is describing in what I've quoted. I wish I had the third book (ch 8–9).
 * In the lead, one can simply mention a generalization to modules without going into specifics. —Quondum 05:06, 26 April 2015 (UTC)


 * You are right. My view was limited because I was coming to the topic through an application's literature. For the projective geometry application, nothing more general than skewfields are needed, so the full generality of the concept is not mentioned. Checking Serge Lang, who follows Bourbaki (an ally if not a member), I see no limitations. My interest in this page is that I want to revamp the Duality (projective geometry) page and I need sesquilinear forms defined over skewfields for that, so this page would be the link for background material. I'd like to see this page modified in the following way: First, the lead needs to be more general in terms of fields and, as you suggest, a nod given to the generalization to modules. Next, the bulk of the current article should be given as an important special case. The section on Baer's work should then be expanded and presented as an application in projective geometry. Finally, a new section dealing with these forms in full generality should be written. I can handle most of this myself, except for this last section, which would be better if done by someone else. Does this sound reasonable? Am I leaving anything out? Bill Cherowitzo (talk) 17:25, 27 April 2015 (UTC)


 * In my (admittedly very limited) perspective, this sounds like a very good way to go about it (sesquilinear forms usually only arise through their applications). This is once case where the generalization seems to be incidental. —Quondum 21:00, 27 April 2015 (UTC)


 * I've written a new lead but I'm not ready to go live with it. I need to do some extra editing in the next couple of sections to make it work properly. Take a look. I think it captures what I said above. Bill Cherowitzo (talk) 19:12, 28 April 2015 (UTC)


 * I'd suggest removing the initial mention of an inner product. It is a poor point of departure (its definition seems vague in WP: it is used in inconsistent ways), and it includes all sorts of structure not relevant here (various relations/constraints). Simply starting at a bilinear form seems appropriate.  However, the general approach looks good.
 * A separate point: the section §Generalization section says "When α is the identity, then f is a bilinear form". This is not true of the exact preceding expression in the case of a noncommutative ring. You also have to move the scalar multiplier on the second parameter to being on the right: f(tx, y) = tf(x, y), f(x, yt) = f(x, y)t. So, technically, a sesquilinear form is not a generalization of a bilinear form in the noncommutative case, more a variant. However, (OR alert) a sequilinear form is equivalent to a bilinear form on a vector space and its "opposite" via a mapping of the second parameter; it is a matter of housekeeping. —Quondum 21:27, 28 April 2015 (UTC)


 * Thanks. I see what you mean about inner product and agree. However, I did like the idea of starting with something simple and concrete ... what if I changed it to dot product (or dot product in Euclidean space)? As to the second point, whoever wrote that failed to mention the fact that since α is an anti-automorphism, the only time it can be the identity is if F is commutative (as Baer points out). When I get around to rewriting this section I'll probably only use Baer as a reference and modernize the terminology by using a more current source. Bill Cherowitzo (talk) 23:28, 28 April 2015 (UTC)


 * You presumably mean "simple and familiar" rather than "simple and concrete" ;). Notwithstanding my dislike of the inconsistency of that article (it gives two incompatible definitions without highlighting the incompatibility, but this is not of significance here; either would do), the Euclidean dot product would probably be an ideal prototype to be generalized to a bilinear form and to a sesquilinear form. The second point was introduced as relating to a field.  Anyhow, it'll get cleaned up. —Quondum 04:20, 29 April 2015 (UTC)


 * Ok, I'll go with that. BTW, for Baer, "field" always means skewfield and he uses the locution "commutative field" when he means otherwise. Yet another good reason not to quote Baer directly. Also, I'm thinking of doing something that I would normally frown upon if someone else did it - here is your chance to talk me out of it. In the math physics literature the second argument is usually taken to be the linear one while in the math literature I've seen a preponderance of first argument linear. I'm thinking of keeping the notation as is for the complex case, but switching (with sufficiently many warnings) to the other for the rest of the article. This I think would give a reader a fighting chance of not getting confused if they head out to the literature (but increase the chances of getting confused if they only skim our article). Bill Cherowitzo (talk) 20:15, 29 April 2015 (UTC)


 * Ah, okay, the field/skewfield thing got me. Who, me – talk you out of my POV preference? You might want to put it as: the convention for this article is linear in the first argument, but in physics the convention is usually linear in the second, i.e. treat the complex case as a brief departure from the convention of the article, rather than introducing the complex case and then switching conventions. I hope I haven't muddled what you meant. I think that the general case (with noncommutativity) being covered in an article trumps any convention used in the commutative case. WRT Baer and noncommutativity, your earlier mention of his handling of the bilinear case suggests that he missed an opportunity to generalize bilinearity in a natural fashion through allowing the two parameters to be different (but closely related) modules, as Bourbaki does. —Quondum 22:36, 29 April 2015 (UTC)


 * No muddling at all, that's exactly what I meant and I'll start revising tomorrow. In defense of Baer, his intentions were to put projective geometry on a firm algebraic footing. If he didn't need something he didn't generalize in that direction. I suspect that he wasn't a big Bourbaki fan. :^) Bill Cherowitzo (talk) 04:03, 30 April 2015 (UTC)

Is every desargesian projective geometry necessarily over a division ring?
The suggestion is that there is a theorem than all desarguesian projective geometries may be constructed as spaces over division rings. Is this true? Desargues's theorem indicates that it is a sufficient condition in the last paragraph of the lead, but not that it is a necessary condition, though it also says later that In dimension 2 the planes for which it holds are called Desarguesian planes and are the same as the planes that can be given coordinates over a division ring. One should also consider the possibility that the axioms of projective geometry get tweaked (e.g. ), so a clarifying qualification of some sort would probably be helpful to the reader. Perhaps something along the lines of "... since every desarguesian projective geometries may be coordinatized by a division ring." If this is unambiguously true, of course. —Quondum 21:03, 3 September 2016 (UTC)
 * For reference, the edit comment As the statement concerned Desarguesian spaces, this was full generality. Perhaps the sentence needs to be reworked to emphasize this.
 * Yes, the desarguesian property is equivalent to being coordinatized by a skewfield. This is a classical result in projective geometry. You can define projective spaces over more general algebraic structures, but they won't be desarguesian in general. I had been thinking that the sentence needed to be modified along the lines you have suggested. Bill Cherowitzo (talk) 04:23, 4 September 2016 (UTC)
 * I've heard of this before, but I do not see it made clear anywhere in WP (maybe just my bad scanning), and it is a significant point that should be in WP. I also find it surprising: it suggests that a division rings might be formalizable as a category, and consequently also fields (I interpret axioms on categories to be constrained to universally quantified equalities, so exclusion of zero divisors and 1≠0 are not permitted), but I thought that this could not be done. I'm also wondering whether the uniqueness of division rings in this regard is sensitive to the exact axiomatization of a desarguesian projective geometry.  —Quondum 06:10, 4 September 2016 (UTC)
 * This is well over my head, but browsing (e.g. Projective Planes by Hughes and Piper) suggests that a particular type of transitivity on the geometry might need to be assumed to imply a division ring. Perhaps an interesting topic for a future investigation. But this does still seem to leave the question open (at least in my muddle-headed thinking).  —Quondum 16:17, 5 September 2016 (UTC)
 * Okay, looking at Baer I think the standard axiomatization of projective spaces with lots of existence and uniqueness requirements (e.g. a unique line defined by two distinct points) evidently forces the division ring. So I've updated it for context; let's see how that works.  I think the axiomatization to accommodate projective geometries over arbitrary rings should still be regarded as out of scope here. —Quondum 05:43, 6 September 2016 (UTC)

Geometric motivation
The section provides (IMO) an exceedingly weak motivation for a sesquilinear form, since it relies on the geometric interpretation on the Argand plane. I would label this as a "geometric interpretation". The fact that it induces a real norm on any complex vector space is a far more compelling as motivation, but does not immediately indicate why such a norm might be of interest. Under the heading of "geometric motivation", the every sesquilinear form corresponds to a correlation (up to a central scalar?) is a far more general and compelling motivation for sesquilinear forms generally from the perspective of projective geometry, and the specialization to symmetric, skew-symmetric, Hermitian and skew-Hermition forms is a direct consequence of the constraint of reflexivity (if one chooses to eliminate the wild automorphisms of C). Seeking reflexive orthogonality on a vector space drives all of this, so one can potentially dispense with the reference to projective geometry and correlations. Opinions on replacement of the section with something with this perspective? —Quondum 23:00, 10 September 2016 (UTC)

Steps of generalization
There are several generalizations of the module used in the definition of sesquilinear forms, depending on field of study, including complex (conjugate-)sesquilinear, vector spaces over general fields, division rings, general modules. This can be very repetitive. Since the case for general modules applies to all other cases essentially without modification, aside from a switch to field-specific terminology, it is tempting to eliminate the repetitiveness. I see the need to provide a separate treatment for the first case since it is a bit much to expect the large number of students of this field to mentally substitute for the unfamiliar terminology, but it is not clear to me that the others need such special treatment (e.g. the division rings of projective geometry could be dropped, especially considering that this field seems to be moving towards the general terminology of modules and to expanding the definition of projective geometries to more general rings). We could provide examples for each of the fields of study. This suggests merging the section on division rings (into which I have already merged the almost identical section for general fields) into that for general rings, with mention of the special role of division rings relating to projective geometry. —Quondum 23:42, 10 September 2016 (UTC)

associated quadratic form
To speak of the associated quadratic form for a sesquilinear form seems like an unnecessary abuse of terminology, especially without definition since it is not a quadratic form and because it seems to be nonstandard. Though I have seen the term in browsing around, I have more often seen the term pseudo-quadratic form in this context, but its definition includes subtleties that I have not mastered. —Quondum 23:54, 10 September 2016 (UTC)

For clarity, the definition that I have seen is that $q(λx) = |λ|^{2}q(x)$. Hardly what I'd consider a fitting use for the term "quadratic". Perhaps we should also call a function a function that satisfies $f(λx) = |λ|f(x)$ "linear"? —Quondum 07:02, 17 September 2016 (UTC)

Viewed as a bilinear form?
The statement
 * A complex sesquilinear form can also be viewed as a complex bilinear map $$\overline{V} \times V \to \mathbf{C} $$ where $\overline{V}$ is the complex conjugate vector space to $V$.

is unreferenced (it appeared in Feb 2007) and appears to be incorrect: I see no way that you can massage a (nonzero) complex sesquilinear map (conjugate linear in one argument and linear in the other) into a bilinear form. I propose removing the statement. —Quondum 06:56, 17 September 2016 (UTC)


 * I believe it is correct. The complex vector space $$\overline{V}$$ has the same underlying abelian group as $$V$$, but the scalar multiplication is redefined so that multiplication by a complex number $$c$$ on $$\overline{V}$$ corresponds to multiplication by $$\bar{c}$$ on $$V$$. Ebony Jackson (talk) 04:33, 27 October 2020 (UTC)

sesquilinear concept is a variant on (not a generalization of) bilinear concept.
In the introduction sesquilinear is described as a "generalization of bilinear", but strictly speaking (pardon the pedantry) this is incorrect. cerniagigante (talk) 07:53, 7 September 2023 (UTC)


 * @Cerniagigante: The definition stated in the article is perhaps a bit suboptimal and not general enough to show that sesquilinear forms generalize bilinear forms. If you say the following:
 * Let K be a field and $$\iota:K\to K$$ a field automorphism that is an involution (i.e. $$\iota\circ\iota=\mathrm{id}$$) and V a vector space over K. Then a map $$T:V\times V\to K$$ is a sesquilinear form if
 * $$T(x+y,v+w)=T(x,v)+T(x,w)+T(y,v)+T(y,w)$$ and $$T(ax,by)=a\iota(b) T(x,y)$$ for every $$x,y,v,w\in V$$ and $$a,b\in K$$
 * then bilinear forms are the special case of $$\iota=\mathrm{id}$$, and so every bilinear form is a sesquilinear form. The definition in the article is for $$K=\mathbb{C}$$ and $$\iota(z)=\overline{z}$$. —Kusma (talk) 08:09, 7 September 2023 (UTC)
 * Thanks. I accept that under this definition, let's call it of a $$\iota$$-sesquilinear form, a bilinear form is a special case as $$\operatorname{id}$$-sesquilinear.
 * But then the usual definition of sesquilinear (which is $$\bar\cdot$$-sesquilinear according to the general definition) is another, different, special case and still not a generalization.
 * What is obvious to a trained professional algebraist may become quite confusing to a casual Wikipedian. Perhaps your clarification could be added further down. cerniagigante (talk) 08:35, 7 September 2023 (UTC)