Talk:Linear fractional transformation

definition of "linear fractional transformation"

This article needs to cite specific sources that define "linear fractional transformation". A Google search found a source that says: "Linear fractional transformations are also called Möbius transformations or bilinear transformations."

• Complex Analysis By Dennis G. Zill, Patrick D. Shanahan (p. 343)

--50.53.240.197 (talk) 01:09, 23 September 2014 (UTC)

Please cite a specific source (with page numbers) for this sentence from the article: "More generally in mathematics, C may be replaced by another ring (A, + , × )." --50.53.240.197 (talk) 01:30, 23 September 2014 (UTC)

According to Google books, "linear fractional transformation" is not mentioned in Dubrovin, et al, volume 1, chapter 2, §15. --50.53.240.197 (talk) 03:26, 23 September 2014 (UTC)

Yaglom mentions "ring" once in a footnote on page 11. --50.53.240.197 (talk) 03:46, 23 September 2014 (UTC)

Since Mobius died before numbers were expanded beyond the complex, use of the eponym "Mobius transformation" for transformations of matrices, functions, operators, continued fractions, or several complex variables expands his importance beyond his work. Therefore the phrase "Linear fractional transformation" is preferred for application of the Mobius-type mapping in these extended domains. The hypercomplex numbers provided an early context for the expanded concept.

A reference to Nicholas John Young has been provided: "Linear fractional transformations in rings and modules". His interest apparently was aroused by application to a companion matrix (Glasgow Mathematics Journal 20(2):129 to 32). A prominent reference is "On linear fractional transformations with operator coefficients"(1974) by M. G. Krein and J.L. Smuljan, appearing in American Mathematical Society Translations 103:125 to 52. More recently (1992) Lawrence Harris contributed "Linear fractional transformations of circular domains in operator spaces", Indiana University Mathematical Journal 41(1):125.

"Mobius transformation" is an ambiguous term as noted in Journal of Geometry and Physics 37(3):183 to 9: sometimes it refers to a linear fractional transformation and sometimes to a homeomorphism generated by reflections through spheres and planes. The descriptive title of this article is preferable to the ambiguous eponym.Rgdboer (talk) 23:16, 23 September 2014 (UTC)

Thanks for the references. The other two references belong in the article, not on the talk page. :-) --50.53.240.60 (talk) 01:20, 24 September 2014 (UTC)

The paper by Harris, "Linear fractional transformations of circular domains in operator spaces", has references to more papers with "linear fractional transformation[s]" in the title:

• [30] R. M. Redheffer, On a certain linear fractional transformation, J. Math. and Physics 39(1960), 269-286.
• [33] Ju. L. Smuljan, On operator balls and linear-fractional transformations with operator coefficients, Math. USSR Sbornik 6(1968), 309-326.
• [34] Ju. L. Smuljan, General linear-fractional transformations of operator balls, Siberian Math. J. 19(1978), 293-298.
• [37] M. Kh. Zakhar-Itkin, The matrix Riccati differential equation and the semi-group of linear fractional transformations, Russian Math. Surveys 28(1973) No. 3, 89-131.

--50.53.240.60 (talk) 01:42, 24 September 2014 (UTC)

To confirm that Hypercomplex numbers featured linear fractional transformations, this article is important:

• Elie Cartan (1908) "Les systems de nombres complex et les groupes de transformations", Encyclopédie des sciences mathématiques pures et appliquées I 1. and Ouvres Completes T.2 pt. 1, pp 107–246.

Cartan uses the "formula" $x'={\frac {ax+b}{cx+d}},$ but mentions neither Mobius nor linear fractional transformations. The formula occurs on pages 452 and 455 discussing quaternions, page 460 for biquaternions, and 465 for Clifford systems.

References to articles that are beyond the capacity of a general reader do not support an article; they give an impression of incomprehensibility. For technical articles more maturity is demanded of the reader. In this case the idea of the transformation is relatively simple: extension of the linear mapping ax + b by introduction of fractions. The only real inhibition is knowledge of the extended concepts of number now in use. Once examples beyond the complex numbers are known, then the transformation idea follows.Rgdboer (talk) 03:38, 24 September 2014 (UTC)

Actually, I was looking for convincing evidence that this article should be called "linear fractional transformation". You have essentially admitted that this is a very advanced topic. IMO, the article should be renamed to something reflecting that, say "linear fractional transformations on spaces other than the complex numbers". You are the expert, so please suggest another title. As it is, there are several redirects with similar names that all point to Möbius transformation:

• Fractional linear transform
• Fractional linear transformation
• Fractional linear transformations
• Fractional-linear map
• Fractional-linear transformation
• Linear fractional map
• Linear fractional transformations

--50.53.240.60 (talk) 05:38, 24 September 2014 (UTC)

It is not a surprise that Cartan did not use "Möbius transformation" nor "transformation fractionnelle linéaire" (the French equivalent of "linear fractional transformation"). As far as I know these terms were not used in French before the second half of 20th century, where they could have been imported from English. The term "homographie" (homography) was preferred, even if it denotes also its multivariate generalization. D.Lazard (talk) 06:51, 24 September 2014 (UTC)

This 1932 review in English of a book by Cartan simply calls the equation $z'={\frac {az+b}{cz+d}}$ a "transformation", although it does mention the term "homographie".

Review of:

• Leçons de Géométrie Projective Complexe, par E. Cartan. Paris, Gauthier-Villars, 1931

in Bull. Amer. Math. Soc., Volume 38, Number 7 (1932)

--50.53.49.58 (talk) 08:23, 24 September 2014 (UTC)

To 50.53.240.60: Thank you for noting the other redirects. Two were made by White Cat Bot while we were discussing. All have now been redirected to this article. As for the title, it is precisely descriptive. The issue is the domain of a function which is not limited complex numbers. Thank you for your interest in improving this encyclopedia. Consider opening a WP:User account.Rgdboer (talk) 21:21, 24 September 2014 (UTC)

You missed this one: Fractional linear transform. --50.53.48.236 (talk) 01:03, 25 September 2014 (UTC)

Done. Thank you.Rgdboer (talk) 01:16, 25 September 2014 (UTC)

Now that you have cut the cord, I would suggest adding an about hatnote, so that users who are expecting to find an article about the familiar transformation on complex variables can get to Möbius transformation without having to read the lead. --50.53.48.236 (talk) 02:36, 25 September 2014 (UTC)

The first line has the Mobius transformation link. It couldn't be higher on the page. Hatnote unnecessary.Rgdboer (talk) 21:42, 25 September 2014 (UTC)

use of the French term "transformation fractionnelle linéaire"

[Comment by D.Lazard copied from the section definition of "linear fractional transformation"]

It is not a surprise that Cartan did not use "Möbius transformation" nor "transformation fractionnelle linéaire" (the French equivalent of "linear fractional transformation"). As far as I know these terms were not used in French before the second half of 20th century, where they could have been imported from English. The term "homographie" (homography) was preferred, even if it denotes also its multivariate generalization. D.Lazard (talk) 06:51, 24 September 2014 (UTC)

This 1955 paper by Sierpiński begins: "Pour établir une propriété paradoxale du segment d'une droite, M. J. von Neumann a démontré l'existence de deux transformations linéaires fractionnaires ..."

• Sierpiński, Wacław. "Sur une relation entre deux substitutions linéaires." Fundamenta Mathematicae 41.1 (1955): 1-5. <http://eudml.org/doc/213341>.

(NB: The quote was copied from the Google search results.)

--50.53.54.109 (talk) 18:37, 25 September 2014 (UTC)

Redheffer star, control theory, scattering

Article expansion request: this article should mention the role of LFT's in control theory and mechanical engineering to represent controller-plant feedback (mechanical vibrations, dead reckoning, alpha-beta filtering, Kalman filtering, etc.) and in scattering theory (particle scattering in quantum field theory, the S-matrix, submarine acoustics, scattering of bound and continuous spectra of diffeq in general). Demonstrate closure under the Redheffer star product of LFT's. Some random PDF's that touch on this incompletely:

• [1] - 16 pages of examples, and defintion of the star product.
• [2] - 16 page powerpoint lecture slides.

Both cover a basic example -- damped harmonic oscillator. Here's another:

• [3] IEEE review
• [4] another nice review.

Some digestion of these reviews deserves a place here. 67.198.37.16 (talk) 19:39, 12 March 2019 (UTC)

It seems that the LFT you are referring to is unrelated (except for the name) with the subject of this article. I may be wrong, but, if there is some relationship, it is unclear from the links you have provided. If I an not wrong, the best is that you write a draft Linear fractional transformation (control theory), and, when it will be ready, to submit it as a new article. D.Lazard (talk) 20:45, 12 March 2019 (UTC)

Huh? Read the current article, as I just left it, with the first few sentences and first formula. An LFT is ... a transformation of the form (ax+b)/(cx+d) with ad-bc not zero. In general, a,b,c,d can be elements of a ring. That's what the original article said, in an utterly opaque and obtuse fashion. In "real life", e.g. in mechanical engineering, the ring is a ring of matrices. Most engineers will not have a clue what a ring is, but they do know what a matrix is and they know basic linear algebra -- so the abstract, formal definition needs to be preceded by a softer, simpler definition. I'm trying to provide that softer, simpler definition, accessible to non-mathematicians. 67.198.37.16 (talk) 21:29, 12 March 2019 (UTC)

I'm not really clear on the details, but think the deal with LFT's in engineering is that they help split out the stable and unstable manifolds of a dynamical system under perturbations. So if you look at the mobius transform article, you'll see the talk of elliptic, parabolic, hyperbolic and loxodromic xforms. Lets ignore loxodromic, for now. (its actually about sailing a ship with fixed angle to the stars, curiously enough; the ship spirals around the globe. I don't know the control-theory analog of this.) The elliptic bits are the orbits of points that "get closer together", or at least "stay together", under the transform. The hyperbolic parts diverge (positive Lyapunov exponent in old-fashioned terminology). The divergent parts create issues for control systems -- aircraft stability, etc. So you ask: what happens if I take some fixed FLT, and replace x by x+delta for some small perturbation delta. How does the system respond? The reason that the quoted articles "look different" to you is that they immediately plug in (x+delta) and then invert the matrix (since the condition ad-bc not zero means the denominator is invertable). The matrices a,b,c,d are just small-time-step transfer matrixs for the time-evolution of the system. or, e.g Kalman filters for rocket-flight control. Gahh. Well, there is also uncertainty in what a,b,c,d are. The deal with the hopf fibration is that you can locally decompose the dynamical system into stable and unstable manifolds. (with hopf fibrations, you "get lucky" and can do it globally; in general, this works only locally). The bit about redheffer star is that it shows you how to compose matrices to get diffeqs with higher-order derivatives in them. Or something like that. I was hunting for a simple explanation of that (I'm not an engineer).

I think engineers will have a pretty clear idea about stable and unstable manifolds, but they won't have a clue what a projective line is, nor will they be able take a ring, in full abstracted glory, and do anything with it. The class of rings is very very big; I have no clue what you can say about stability or factorization for rings in general, but for real-valued or complex-valued invertible matrices representing small-time-step evolution of finite-difference equations ... this can be actually worked with. Concrete things can be said. I'm just splatting all this out because this is my casual, informal understanding; and was hoping to find a more rigorous, more detailed explanation here. (Which I didn't.)67.198.37.16 (talk) 22:27, 12 March 2019 (UTC)