Reflection through the origin
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In mathematics, reflection through the origin refers to the orthogonal transform (Euclidean isometry) of 
, also written − I or scalar multiplication by − 1. In coordinates, in two dimensions, this is the map that sends 
, in three dimensions, this sends 
, and so forth.
The term is an abuse of language: like a reflection, it is an involution, meaning that it is its own inverse, but (except in 1 dimension), it is not reflection through a line, plane, or hyperplane.
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[edit] Representations
As a scalar matrix, it is represented in every basis by a matrix with − 1 on the diagonal, and, together with the identity, is the center of the orthogonal group O(n).
It is a product of n orthogonal reflections (reflection through the axes of any orthogonal basis); note that orthogonal reflections commute.
In 2 dimensions, it is in fact rotation by 180 degrees, and in dimension 2n, it is rotation by 180 degrees in n orthogonal planes;[1] note again that rotations in orthogonal planes commute.
[edit] Properties
It has determinant ( − 1)n (from the representation by a matrix or as a product of reflections). Thus it is orientation-preserving in even dimension, thus an element of the special orthogonal group SO(2n), and it is orientation-reversing in odd dimension, thus not an element of SO(2n+1) and instead providing a splitting of the map 
, showing that 
as an internal direct product.
- Together with the identity, it forms the center of the orthogonal group.
- It preserves every quadratic form, meaning Q( − v) = Q(v), and thus is an element of every indefinite orthogonal group as well.
- It equals the identity if and only if the characteristic is 2.
- It is the longest element of the Coxeter group of signed permutations.
Analogously, it is a longest element of the orthogonal group, with respect to the generating set of reflections: elements of the orthogonal group all have length at most n with respect to the generating set of reflections,[2] and reflection through the origin has length n, though it is not unique in this: other maximal combinations of rotations (and possibly reflections) also have maximal length.
[edit] Geometry
If reflection through the origin is in SO(n), then it is the farthest point from the origin; if it is in the other component, there is no natural sense in which it is a farther point but it does provide a base point in the other component.
[edit] Clifford algebras, Spin groups, and Pin groups
It should not be confused with the element 
in the Spin group. This is particularly confusing for even Spin groups, as 
, and thus in Spin(n) there is both − 1 and 2 lifts of − I.
Reflection through the identity extends to an automorphism of a Clifford algebra, called the main involution or grade involution.
Reflection through the identity lifts to a pseudoscalar.
[edit] See also
[edit] Notes
- ^ "Orthogonal planes" meaning all elements are orthogonal and the planes intersect at 0 only, not that they intersect in a line and have dihedral angle 90°.
- ^ This follows by classifying orthogonal transforms as direct sums of rotations and reflections, which follows from the spectral theorem, for instance.
