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Using Eigenvalue Analysis to Rotate in 3D

The continuous rotation of one right-handed coordinate system into another depends on the axis of rotation, the angle, and the sense of rotation. All these can be obtained from the eigenvalues and vectors of the change of basis matrix. (The matrix inverse gives only the end state of the rotation.) This Demonstration shows the axis of rotation and a slider that governs the rotation.

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In engineering we often want to transform quantities such as forces from one coordinate system into another. So consider an arbitrary right-handed coordinate system relative to a fixed coordinate system. We know the components of the unit vectors , , and along the , , and axes. Then, any vector given in the system can be resolved in the system using simple matrix inversion. That is, if we assemble the matrix of components
and if the vector is given in the system as
,
then the same vector is given in the system by
.
However, matrix inversion only tells us the start and end states of the vector and not the process of rotation from one coordinate system to another, namely, the axis about which the system is rotated to bring it to the system, the angle of rotation , and the sense of the rotation (clockwise or counterclockwise).
The process [1] involves finding the eigenvalues and eigenvectors of . The eigenvector corresponding to the eigenvalue of 1 gives the axis ; it is the only eigenvector whose components are all real. The two other eigenvalues are and , whose eigenvectors are complex.
Therefore, by suitably ordering the eigenvalues and eigenvectors with the real part of the eigenvalues, the vector can be deduced. That is, is the eigenvector of the eigenvalue whose real part is the maximum.
The value is given by
.
Suppose the sense of rotation is defined in the direction of using the value defined as follows:
.
Take one of the complex eigenvectors and let . Then
.
In this Demonstration, a random right-handed coordinate system with gray planes is first generated. (You can generate a new system to see different cases.) The fixed system (red axes and planes) is generated by taking the inverse of the column matrix . Use the slider to continuously rotate the system (blue axes and planes) about the vector (purple) into the system.
Reference
[1] L. Sadun. Rotation in 3 Dimensions [Video]. (Nov 21, 2014) www.youtube.com/watch?v=5BtDR25TE8k.
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