A Note on the Diagonalizability and the Jordan Form of the 4×4 Homogeneous Transformation Matrix

The $4×4$ homogeneous transformation matrix is extensively used for representing rigid body displacement in 3D space and has been extensively used in the analysis of mechanisms, serial and parallel manipulators, and in the field of geometric modeling and computed aided design. The properties of the transformation matrix are very well known. One of the well known properties is that a general $4×4$ homogeneous transformation matrix cannot be diagonalized, and at best can be reduced to a Jordan form. In this paper, we show that the $4×4$ homogeneous transformation matrix can be diagonalized if and only if displacement along the screw axis is zero. For the general transformation with nonzero displacement along the axis, we present an explicit expression for the fourth basis vector of the Jordan basis. We also present a variant of the Jordan form which contains the motion variables along and about the screw axis and the corresponding basis vectors which contains the information only about the screw axis and its location. We present a novel expression for a point on the screw axis closest to the origin, which is then used to form a simple choice of basis for different forms. Finally, the theoretical results are illustrated with a numerical example.