  Rotations and poses are ubiquitous throughout many fields of science and
engineering such as robotics, aerospace, computer vision and graphics. In this
paper, we provide a complete characterization of rotations and poses in terms
of the eigenstructure of their matrix Lie group representations, SO(3), SE(3)
and Ad(SE(3)). An eigendecomposition of the pose representations reveals that
they can be cast into a form very similar to that of rotations although the
structure of the former can vary depending on the relative nature of the
translation and rotation involved. Understanding the eigenstructure of these
important quantities has merit in and of itself but it is also essential to
appreciating such practical results as the minimal polynomial for rotations and
poses and the calculation of Jacobians; moreover, we can speak of a
principal-axis pose in much the same manner that we can of a principal-axis
rotation.
