Using Eigenvalue Analysis to Rotate in 3D

Requires a Wolfram Notebook System


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.

Contributed by: Raja Kountanya (December 2014)
Open content licensed under CC BY-NC-SA





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.


[1] L. Sadun. Rotation in 3 Dimensions [Video]. (Nov 21, 2014)

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