Isometry of space/Rotation axis/Complement/Unitary/Section


Let

be a linear isometry. Then there exists an eigenvector with eigenvalue

or .

The characteristic polynomial of is a normed polynomial of degree three. For , we have , and for , we have . Due to the intermediate value theorem, has a real zero. Such a zero is an eigenvalue of . Because of fact, this eigenvalue equals or .


A proper linear isometry of the space transforms the unit sphere into itself. One might imagine an isometry as a rotation of a ball lying in a suitable bowl.


A proper isometry

has an

eigenvector with eigenvalue , that is, there exists a line (through the origin)

that is a fixed line for .

We consider the characteristic polynomial of , that is,

This is a normed real polynomial of degree three. For , we have

As the polynomial as , there must be a positive that is a zero of . Due to fact, we must have .



Let

be a proper isometry. Then is a rotation around a fixed axis. This means that is described, with respect to a suitable orthonormal basis, by a matrix of the form

Due to fact, there exists an eigenvector for the eigenvalue . Let denote the corresponding line. This line is fixed and, in particular, invariant under . Because of fact, also the orthogonal complement is invariant under . That is, there exists a linear isometry

that coincides on with . Here, is proper. Therefore, by fact, is a plane rotation. If we choose a vector of length one in , and an orthonormal basis of , then has with respect to this basis the given matrix form.