# Retrieving angle of rotation from transformation matrix

by Phong
Tags: angle, matrix, retrieving, rotation, transformation
 P: 5 Hi! How do I calculate the angle of rotation for each axis by a given 4x4 transformation matrix? The thing is that all values are a kind of mixed up in the matrix, so I cannot get discrete values to start calculating with anymore. Thanks, Phong
 Math Emeritus Sci Advisor Thanks PF Gold P: 38,882 A four by four transformation matrix? Are you rotating in four dimensional space or is this a projective space? First find the eigenvalues. A rotation matrix, in four dimensions may have two real and two complex-conjugate eigenvalues or two pairs of complex eigenvalues. If there are two real eigenvalues they must be either 1 or negative one. The eigenvectors corresponding to those eigenvalues give the axes of rotation. The complex eigenvalues will have modulus 1 and are of the form $cos(\theta)\pm i sin(\theta)$ where $\theta$ is the angle of rotation. Two pairs of complex rotation give two simultaneous rotations in four space but are again of the form $cos(\theta)+ i sin(\theta)$. What those mean depends upon how you are writing vectors in four space. If you are talking about a matrix representing a rotation matrix projectively, then you can renormalize to make the last row [0 0 0 1] and the last column $$\begin{bmatrix}0 \\ 0 \\ 0\\ 1\end{bmatrix}$$. The 3 by 3 matrix made up of the first three rows and columns will have one eigenvalue of 1 (the corresponding eigenvector gives the axis of rotation) and two complex conjugate eigenvalues of modulus 1. They will be of the form $cos(\theta)+ i sin(\theta)$ where $\theta$ is the angle of rotation.
 P: 5 Hello! Thank you very much for your detailed reply. I must admit that I'm pretty new to transformation matrices and have not yet entirely understood the mathematical meaning of eigenvalues and eigenvectors although I try hard to understand everything I can read about it, but with some help I surely learn a lot faster. I'm rotating in a projective three-dimensional space, that's why I use a 4x4 matrix. To give a more specific example, I have a transformation matrix that is the following: $$\begin{bmatrix}0.893 & 0.060 & -0.447 & 20 \\ -0.157 & 0.97 & -0.184 & 15 \\ -0.423 & -0.235 & -0.875 & 45 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ This transformation matrix should transform the object with a translation of 20 15 45 and a rotation of -15 25 -10 (xyz). Now the eigenvalues. I don't know if I've understood the meaning of them correctly, but if yes the eigenvalues for this matrix should be in the identity matrix which is: $$\begin{bmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ So if I'm still on the right track, the eigenvectors, which are the axes of rotation, are simply $$\left(1, 0, 0, 0)$$ for x $$\left(0, 1, 0, 0)$$ for y $$\left(0, 0, 1, 0)$$ for z Am I still on the right track or am I totally and fatally wrong?

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