Talk:Conformal linear transformation

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✠ SunDawn ✠ (contact) 04:43, 16 January 2024 (UTC)[reply]

Meaning of words

“The basis matrix” is a nonstandard, unexplained phrase. “The basis” needs a referent to have any chance of making sense (whence cometh a basis? this list is allegedly a list of properties of a particular kind of matrix) but I suspect the problem goes deeper. Is this just a pile of WP:OR? Please find a decent source and use it to replace this incoherent list with something that uses standard terminology correctly. 100.36.106.199 (talk) 12:34, 20 January 2024 (UTC)[reply]

"Basis" is standard terminology in linear algebra. See Basis (linear algebra), which is linked on the page in the very sentence you complain about. A basis matrix is a matrix that is a basis. Please read up on the subject before proposing that something is wrong with the page. Aaronfranke (talk) 06:24, 5 February 2024 (UTC)[reply]

The phrase “a basis matrix is a matrix that is a basis” is meaningless gibberish. I am sorry that your grasp of linear algebra is so poor that you do not understand that a rectangular array of numbers and a set of elements of a vector space are different kinds of objects, but really until you recognize the incoherence of what you’ve written (e.g., by comparing it directly to what reliable sources say) you should not be editing this article (or any other article on mathematics). 100.36.106.199 (talk) 13:02, 6 February 2024 (UTC)[reply]

I have reported your behavior here: https://en.wikipedia.org/wiki/Wikipedia:Administrators%27_noticeboard/Edit_warring#User:100.36.106.199_reported_by_User:Aaronfranke_(Result:_) Aaronfranke (talk) 08:09, 8 February 2024 (UTC)[reply]

For what it's worth, I think this report was inappropriate, both of you were violating policy by revert warring instead of discussing, both of you were using inappropriately aggressive language (I do this also sometimes, so this is not intended too harshly) and the IP editor should be unbanned. The IP editor's concerns were/are valid even if their language was dismissive/insulting. –jacobolus (t) 20:11, 8 February 2024 (UTC)[reply]

I too am confused by the term "basis matrix", despite knowing linear algebra including bases, matrices, change of basis, etc. Does it mean that the columns of the matrix form a basis? But apparently squareness is also required. So the matrix is invertible? Mgnbar (talk) 12:31, 8 February 2024 (UTC)[reply]

Also I don't understand the other phrases under dispute, such as "The basis may be composed of rotation." Please clarify all of the content that 100.36.106.199 was complaining about. Mgnbar (talk) 14:00, 8 February 2024 (UTC)[reply]

Presumably he means that the transformation can be expressed as a composition of dilation, reflection and rotation. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:26, 8 February 2024 (UTC)[reply]

The phrase "a basis matrix" makes sense, but the phrase "the basis matrix" does not. Earlier, the phrase high-level transform decomposition has no obvious meaning. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:26, 8 February 2024 (UTC)[reply]

Sorry, but what is "a basis matrix" even? Do you mean a matrix that can be a member of a basis for the (n m)-dimensional vector space of m x n matrices? That would be any non-zero m x n matrix. So you must mean something else. Do you mean a change-of-basis matrix? Mgnbar (talk) 16:32, 8 February 2024 (UTC)[reply]

Presumably a transformation from a preferred basis to a different basis, but even if my guess is correct the text should spell it out. If he doesn't mean a change of basis matrix then I'm completely in the dark. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:26, 8 February 2024 (UTC)[reply]

Name of the concept

This concept is valuable and under-applied, but I don't think the name "conformal linear transformation" is a good one for it. The name seems relatively rare in practice and not really in currency (edit: after searching there doesn't seem to be any particularly common name for this). I don't think primarily focusing on these transformations being "conformal" is at all helpful, since the geometry of >2-dimensional conformal transformations (Möbius transformations) is entirely different, and the 2-dimensional case (where the concept of "conformal" originated) dramatically more different still. What about a name like Tristan Needham's "amplitwist" or similar? If I needed a descriptive phrase I would go for something like "origin preserving similarity transformation". A few sources seem to use the names "homogeneous similitude" or "homogeneous similarity transformation", which seem like okay names.

One of these transformations is most naturally represented using geometric algebra, as a versor (product of invertible vectors). Sandwich multiplying a vector $v$ by a versor $A$ like $AvA^{\dagger }$ where $A^{\dagger }$ is the reverse of $A$ applies the transformation. So hunting for sources discussing this may turn up other names for the concept. The group of transformations is called the versor group or Lipschitz group. –jacobolus (t) 17:12, 8 February 2024 (UTC)[reply]

I'll note that we have the article conformal group, which is also not in a good state. (Also the article Möbius group is limited to two dimensions.) I'm not convinced the present article needs to exist, but some clarity in adjacent articles seems more pressing. Tito Omburo (talk) 00:29, 9 February 2024 (UTC)[reply]

Neither conformal group nor Möbius group (which are as far as I can tell the same thing) is the same as these orthogonal-transformation-composed-with-dilation transformations. I think this concept is worth having a separate article about, but it should be clear and clearly sourced.

The Möbius transformation article is very problematic because it mixes up the geometric transformations generated by reflections and sphere inversions (the actual subject of the term "Möbius transformation") with the particular representation of 2-dimensional Möbius transformations as fractional linear transformations of the complex projective line. I'd like to eventually fix this but it's going to take significant work. –jacobolus (t) 00:38, 9 February 2024 (UTC)[reply]

First definition in the conformal group article defines the "conformal orthogonal group" to be as in this article, but there is a lack of clarity as to the subject there, which overlaps with this article and Möbius. Tito Omburo (talk) 01:18, 9 February 2024 (UTC)[reply]

I don't thin "conformal orthogonal group" should be covered at the article conformal group. It's a big mistake to mash two largely unrelated topics together. –jacobolus (t) 01:29, 9 February 2024 (UTC)[reply]