Talk:Sesquilinear form

message: 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)
thread_title:  Standard Hemitian form 

message: 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)
replies: text: 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)
thread_title:  The first argument should be linear 

message: 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)
thread_title:  Equivariance 

message: 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)
replies: text: 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)
thread_title:  "Standard" for mathematicians 

message: 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)
replies: text: 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)
thread_title:  Too many hyperlinks!!! 

message: *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.
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)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
thread_title:  Is every desargesian projective geometry necessarily over a division ring? 

message: 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)
thread_title:  Geometric motivation 

message: 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)
thread_title:  Steps of generalization 

message: 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)
thread_title:  associated quadratic form 

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