Recall that an affine map f(x) = A x + b from R^{n} to R^{n} is an isometry if and only if the matrix A is an orthogonal matrix.
Suppose that f(x) = U x + b is an isometry of the plane (with U a 2 \times 2 orthogonal matrix).
If the matrix 1-U has rank 2, then f is a rotation, and the center of rotation is (1-U)^{-1}b.
If the matrix 1-U has rank 1, then f is a reflection if b is parallel to a non-zero column of the matrix 1-U and a glide reflection for any other non-zero value of b.
If the matrix 1-U has rank 0, then f is a translation.
If an isometry f of the plane is a rotation about the point p, then for every point x in the plane the rotational center p must lie on the perpendicular bisector of the line segment from x to f(x).
Let f be the rotation of the plane about the point (-1, 3) counterclockwise through the angle 2pi/3. Find a matrix A and a vector b such that such that f(x) = A x + b for all points x in the plane.
Let M be the matrix
| cos theta | - sin theta |
| sin theta | cos theta |
where theta is not an integer multiple of 2pi. Explain why the affine map f given by f(x) = M x + b must, for every value of the vector b, be the rotation about some point in the plane through the angle theta.
Use the result stated in the previous exercise to prove that the composition of two rotations, regardless of whether they have the same center, is always a rotation whenever the sum of the two angles of rotation is not an integer multiple of 2pi.
Show that the only possible real eigenvalues of an orthogonal matrix are 1 and -1.