Involutions in projective geometry, part I.
20 Feb 2023I am reading Richter-Gebert’s book (1, Richter-Gebert, 2011) and I’m in the section on Involutions and Quadrilateral Sets (chapter 8).
The involution discussion uses an example of projective transformations on a line: \(\tau: \mathbb{R}\mathbb{P}^1 \mapsto \mathbb{R}\mathbb{P}^1\). Within the book, \(\mathbb{R}\mathbb{P}^1\) is the Real Projective 1-dimensional space. Homogenous coordinates in \(\mathbb{R}\mathbb{P}^1\) are of the form \(\left [ x, w \right]\).
An involution is “let’s apply a function to a point and then apply the same function again to that output. If the resulting point after applying the function twice is the initial point, we have an involution.” In other words, where \(\tau\) is a function with one argument, and \(p \in \mathbb{R}\mathbb{P}^1\):
\[\tau(\tau(p)) = p\]If \(\tau\) is a matrix \(\mathbf{T}\), then \(\mathbf{T}^2= \lambda I_d\), where \(I_d\) is the identity matrix for dimension \(d\). For the example of \(\mathbb{R}\mathbb{P}^1\), \(\mathbf{T}\) is a \(2 \times 2\) matrix.
So we can also write,
\[\mathbf{T}(\mathbf{T}(p)) = p\]The requirement that \(\mathbf{T}^2= \lambda I_d\) means there’s only a few options for \(\lambda\) in this \(\mathbb{R}\mathbb{P}^1\) space.
Apply the determinant to both the left and right sides to get,
\[\det(\mathbf{T}\mathbf{T}) = \det(\lambda I_d)\]using the property that the determinant of a matrix product is the product of the determinants of the individual matrices,
\[\det(\mathbf{T})\det(\mathbf{T}) = \det(\lambda I_d)\]The determinant of \(\lambda \left[\begin{array}{c c} 1 & 0\\ 0 & 1 \end{array}\right]\) is \(\lambda^2\), so substituting on the right,
\[\det(\mathbf{T})\det(\mathbf{T}) = \lambda^2\]From this \(\sqrt{\lambda^2} = -\lambda \; \text{or} \; \lambda\) (again, in the \(\mathbb{R}\mathbb{P}^1\) case), so \(\det(\mathbf{T})\) is either \(-\lambda\) or \(\lambda\).
What are the options for \(\lambda\)? Very few – either \(+1\) or \(-1\).
Other reference:
Wikipedia: The section on Euclidean geometry, and below are sections on involutions in Projective geometry and Linear Algebra.
Cited references
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[1] J. Richter-Gebert, Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, 1st ed. Springer Publishing Company, Incorporated, 2011.