Here, we consider rotations in R³. Consider that you are an engineer needing to program the motions of a robotic arm. The arm has a fixed legth of 1 meter, and two directions of rotational pivot. In the following, imagine the robotic arm as a unit vector in R³. The first direction of rotation is in the x - y plane. A rotation by an angle in the x - y plane is obtained by applying the following matrix to a vector Ray (0): Ryz (0)= (cos() sin (0) 0 = The second direction of rotation is in the y - z plane. A rotation by an angle o in the y - z plane is obtained by applying the following matrix to a vector ( 9) sin(0) 0 cos(0) 0 with similar constraints on 1, 2 and 0. 0 0 cos(p) - sin(0) sin() cos(p) The robotic arm can only move by combinations Ryz (o) and Ray (0), but, we can perform any combination of these in succession. But, with some practice we discover that every required motion can be obtained by either A = Ray (02) Ryz (0) Ray (01), for some angles 01, 02, and all between 0° and 180°, or by B = = Ryz(02) Ray (0) Ryz (01), Our overall goal is to understand the eigenvalues and eigenvectors of A and B, but to do so directly is computationally very difficult. So, we'll dance around direct computations and ask questions related to this. Theoretically, A and B have the exact same properties (just change the labels on the axes and the positions of the coordinates), so it suffices to study A alone.