On the Product of Coninvolutory Affine Transformations

TL;DR

This paper investigates how affine transformations over complex numbers can be decomposed into products of coninvolutions, providing necessary and sufficient conditions and characterizations for such decompositions.

Contribution

It characterizes affine transformations as products of coninvolutions based on linear part properties and introduces bounds on the number of coninvolutions needed for decomposition.

Findings

Affine transformation $g$ is a product of two coninvolutions iff $L(g)$ is $c$-reversible.

Elements can be expressed as products of three coninvolutions via consimilarity.

Transformations with $| ext{det}(A)|=1$ are products of at most four coninvolutions.

Abstract

A complex matrix is called \emph{coninvolutory} if $T\overline{T}=I$. In this paper, we study decompositions of affine transformations in $\mathrm{Aff}(n,\mathbb{C})=\mathrm{GL}(n,\mathbb{C})\ltimes \mathbb{C}^n$ into products of coninvolutions. We prove that an affine transformation $g$ is a product of two coninvolutions in $\mathrm{Aff}(n,\mathbb{C})$ if and only if its linear part $L(g)$ is $c$-reversible; that is, $L(g)$ is conjugate to $\overline{L(g)}^{-1}$ in $\mathrm{GL}(n,\mathbb{C})$. Equivalently, $g$ is conjugate to $\overline{g}^{-1}$ in $\mathrm{Aff}(n,\mathbb{C})$. We further characterize elements that are products of three coninvolutions via consimilarity and show that every $g=(A,v)\in \mathrm{Aff}(n,\mathbb{C})$ with $|\det(A)|=1$ can be expressed as a product of at most four coninvolutions.