Talk:Sesquilinear form

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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[edit]

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[edit]

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)

Equivariance[edit]

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[edit]

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. — Carl (CBM · talk) 13:24, 5 November 2013 (UTC)

Too many hyperlinks!!![edit]

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

Generalization to modules[edit]

"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)