linear algebra show an orthogonal map of the plane is either

linear algebra

show an orthogonal map of the plane is either a reflection or a rotation

Solution

An orthogonal transformation of R³ is either a rotation or the composition of rotation and reflection about a plane depending on when its determinant is +1 or -1.

Let an element T of O(3) has an eigenvalues which is eithe +1 or -1. If = +1 then T has a fixed vector a and acts like an element of O(2) in the plane perpendicular to a.

If it acts a rotation here then we have rotation in R³ .

If it acts as a reflection in a vector b here, then we have a reflection in the plane containing a and b.

If T has no positive eigen values than as above we have = -1 with eigenvector c. As before T acts as an element of O(2) in the plane perpendicular to c and it can not act as a reflection here otherwise it would have a fixed vector (which would be an eigen vector + 1.) Hence it act as a rotation in this plane.

(1) The perpendicular of two reflection (which have determinant -1) has determinant +1 and so is a rotation. Since vector b is fixed and it is a rotation about b.

(2) Rotation by about the z-axis followed by reflection in the (x, y)-plane.

(3) Product of rotation is always rotation since O(3) is a subgroup . Seeing what the axis is can be rather tricky. For example, rotation (by /2 anticlockwise) about the x-axis, followed by rotation about the y-axis followed by rotation about the z-axis is rotation by /2 about the y-axis