Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1

• Oct 14th 2013, 07:45 PM
abscissa
Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1
Hi,
I have a question about describing geometrically the action of an arbitrary orthogonal 3x3 matrix with determinant -1. I would like to know if my proposed solutions are satisfactory, or if they lack justification. I have two alternate solutions, but have little confidence in their validity. Any help would be greatly appreciated!!

Solution 1: The orthogonal 3 x 3 matrix with determinant 1 is an improper rotation, meaning it is a reflection combined with a proper rotation. In another sense, an improper rotation is an indirect isometry, which is an affine transformation with an orthogonal matrix with a determinant 1.

Solution 2:
A rotation about the origin, followed by inversion through the origin (i.e. (x,y,z)-->(-x,-y,-z) ). Note that a "left-handed object" turns into a "right handed object", so "handedness is reversed" but otherwise it is just like a rotation.