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TECHNICAL BRIEFS

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

[+] Author and Article Information
Sangamesh Deepak R

Department of Mechanical Engineering, IISc Bangalore, Indiasangu.09@gmail.com

Ashitava Ghosal1

Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560 012, Indiaasitava@mecheng.iisc.ernet.in

Unless stated otherwise, we assume $[R]$ to be nontrivial, i.e, $[R]≠[I3]$.

It may be noted that since $[R]$ is real, the eigenvectors $x1$ and $x2$ are related by $x2=ax¯1$, for some $a∊C$, where $x¯1$ is complex conjugate of $x1$.

Four-dimensional vectors are indicated in boldface lower-case and by using a tilde.

The symbols $x1$, $x2$, $x3$, and $x4$ are the coordinates of a vector in $R4$, with respect to standard ordered basis $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, and $(0,0,0,1)$.

The quantities $x1$ and $x2$ that appear in $f̃1$ and $f̃2$ could always be found if $k$ is known as illustrated in the example of Sec. 4.

1

Corresponding author.

J. Mech. Des 128(6), 1343-1348 (Dec 19, 2005) (6 pages) doi:10.1115/1.2338579 History: Received June 29, 2005; Revised December 19, 2005

## Abstract

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.

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## Figures

Figure 1

Geometrical construction to derive r0

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