# Affine involution

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{{short description|Linear or affine transformation which is its own inverse}}
{{more references |date=August 2022}}
{{Use dmy dates|date=October 2025}}

In [Euclidean geometry](/source/Euclidean_geometry), an '''affine involution''' is an [involution](/source/Involution_(mathematics)) which is a [linear](/source/linear_transformation) or [affine](/source/affine_transformation) transformation over the [Euclidean space](/source/Euclidean_space) {{tmath|\R^n}}. Such involutions are easy to characterize and they can be described geometrically.<ref>{{Cite book |last=Books LLC |url=https://books.google.com/books?id=TJuISQAACAAJ |title=Affine Geometry: Affine Transformation, Hyperplane, Ceva's Theorem, Affine Curvature, Barycentric Coordinates, Centroid, Affine Space |publisher=[General Books LLC](/source/General_Books_LLC) |year=2010 |isbn=978-1-155-31393-1 |ol=OL60673682M |archive-url=https://web.archive.org/web/20251120025524/https://books.google.com/books?id=TJuISQAACAAJ |archive-date=2025-11-20 |url-status=live}}</ref>{{Clarify|reason=Too brief; almost irrelevant.|date=April 2026}}

==Linear involutions==
To give a linear involution is the same as giving an [involutory matrix](/source/involutory_matrix), a [square matrix](/source/square_matrix) {{math|'''A'''}} such that
<math display=block>\bold A^2 =  \bold I \quad\quad\quad\quad (1)</math>
where {{math|'''I'''}} is the [identity matrix](/source/identity_matrix).<ref>{{Cite web
| last        = Weisstein
| first       = Eric W.
| title       = Involutory Matrix
| url = https://mathworld.wolfram.com/InvolutoryMatrix.html 
| access-date = 20 October 2025 
| website     = mathworld.wolfram.com 
| language    = en
}}</ref>

It is a quick check that a square matrix {{math|'''D'''}} whose elements are all zero off the main diagonal and ±1 on the diagonal, that is, a [signature matrix](/source/signature_matrix) of the form

<math display=block>\bold D = \begin{pmatrix}
\pm 1   & 0       & \cdots & 0       & 0      \\
0       & \pm 1   & \cdots & 0       & 0      \\
\vdots  & \vdots  & \ddots & \vdots  & \vdots \\
0       & 0       & \cdots & \pm 1   & 0      \\
0       & 0       & \cdots & 0       & \pm 1  
\end{pmatrix}</math>

satisfies (1), i.e. is the matrix of a linear involution. It turns out that all the matrices satisfying (1) are of the form 
<math display=block>\bold A =  \bold U^{-1} \bold{DU},</math>
where {{math|'''U'''}} is invertible and {{math|'''D'''}} is as above. That is to say, the matrix of any linear involution is of the form {{math|'''D'''}} [up to](/source/up_to) a [matrix similarity](/source/matrix_similarity). Geometrically this means that any linear involution can be obtained by taking [oblique reflection](/source/oblique_reflection)s against any number from 0 through {{mvar|n}} [hyperplane](/source/hyperplane)s going through the origin. (The term ''oblique reflection'' as used here includes ordinary reflections.)

One can easily verify that {{math|'''A'''}} represents a linear involution if and only if {{math|'''A'''}} has the form
<math display=block>\bold A = \pm (2 \bold P - \bold I)</math>
for a linear [projection](/source/Projection_(linear_algebra)) {{math|'''P'''}}.

==Affine involutions==

If ''A'' represents a linear involution, then ''x''→''A''(''x''−''b'')+''b'' is an [affine](/source/affine_transformation) involution. One can check that any affine involution in fact has this form. Geometrically, this means that any affine involution can be obtained by taking oblique reflections against any number from 0 through ''n'' hyperplanes going through a point ''b''.<ref>{{Cite web |last1=Bamberg |first1=John |last2=Penttila |first2=Tim |title=Analytic Projective Geometry |url=https://files.znu.edu.ua/files/Bibliobooks/Inshi83/0062408.pdf |url-status=live |archive-url=https://web.archive.org/web/20251021234003/https://files.znu.edu.ua/files/Bibliobooks/Inshi83/0062408.pdf |website=Zaporizhia National University |archive-date=21 October 2025 }}</ref> 

Affine involutions can be categorized by the dimension of the [affine space](/source/affine_space) of [fixed point](/source/Fixed_point_(mathematics))s; this corresponds to the number of values 1 on the diagonal of the similar matrix ''D'' (see above), i.e., the dimension of the [eigenspace](/source/eigenspace) for [eigenvalue](/source/eigenvalue) 1.

The affine involutions in <math>}\mathbb{R^3</math> are:<ref>{{Cite journal |last1=Marberg |first1=Eric |last2=Zhang |first2=Yifeng |date=March 2022 |title=Affine transitions for involution Stanley symmetric functions |journal=European Journal of Combinatorics |volume=101 |article-number=103463 |doi=10.1016/j.ejc.2021.103463|arxiv=1812.04880 |s2cid=119290424 }}</ref>
* the [identity](/source/Identity_function)
* the [reflection](/source/Point_reflection) in respect to a point
* the [oblique reflection](/source/oblique_reflection) in respect to a line
* the [oblique reflection](/source/oblique_reflection) in respect to a plane

==Isometric involutions==
In the case that the eigenspace for eigenvalue 1 is the [orthogonal complement](/source/orthogonal_complement) of that for eigenvalue −1, i.e., every eigenvector with eigenvalue 1 is [orthogonal](/source/orthogonal) to every eigenvector with eigenvalue −1, such an affine involution is an [isometry](/source/isometry). The two extreme cases for which this always applies are the [identity function](/source/identity_function) and [inversion in a point](/source/inversion_in_a_point).

The other involutive isometries are inversion in a line (in 2D, 3D, and up; this is in 2D a [reflection](/source/reflection_(mathematics)), and in 3D a [rotation](/source/rotation) about the line by 180°), inversion in a plane (in 3D and up; in 3D this is a reflection in a plane), inversion in a 3D space (in 3D: the identity), etc.

== References ==
{{reflist}}

{{DEFAULTSORT:Affine Involution}}
Category:Affine geometry

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Adapted from the Wikipedia article [Affine involution](https://en.wikipedia.org/wiki/Affine_involution) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Affine_involution?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.