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Eigen-properties of a Rotation Matrix

07/10/2011

Let R be a rotation matrix R^TR=I and \mathrm{det} R=1. Usually we don’t talk about rotation matrix in higher space, though the definition of the rotation matrix is not constrained on 2D or 3D.

1. Eigenvalues and Eigenvectors:

Consider the rotation axis of R is x, then Rx=x. So R must have an eigenvalue as 1. The corresponding eigenvector is the rotation axis.

The other eigenvalues are all complex. And |\lambda_i|=1. That means the eigenvalues is the root of \lambda^d=1. For 3D, the eigenvalues are

\{1, e^{-\theta}, e^{+\theta}\}

2. Singular values and Singular vectors:

An SVD of A is A=U\Sigma V^T. The singular values of A is the square root of the eigenvalues of A^TA and AA^T. If A is a rotation matrix, then A^TA=AA^T=I. So the eigenvalues of I is 1. So the singular values of A is all 1. So the condition number of a rotation matrix is

\kappa=\frac{\lambda_{max}}{\lambda_{min}}=1

So the numerical stability of R is the best!

The singular vectors of A is respectively the eigenvectors of A^TA=I and AA^T=I.

Questions remained:

(1) why does the eigenvalues of R be the root of x^d=1?

(2) the eigenvector of R associating its complex eigenvalue is real or complex?

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