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There's some material briefly mentioned the eigenvectors and eigenvalues when it comes to rotation matrices.

Can someone give me a neat explanation for what eigenvectors and eigenvalues do in rotation operations?

niebayes
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  • You probably need to be more specific here. Do you mean real eigenvalues or complex ones? – Karl Jun 13 '20 at 08:44

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In $\Bbb R^2$ a rotation matrix ($\neq$ identity and reflection) does not have any eigenvectors/eigenvalues in $\Bbb R$, because there is no direction, which is fixed by rotation.

In $\Bbb R^3$ a rotation matrix has the eigenvalue 1 and the corresponding eigenvector spans the axis of rotation. Again there are (up to the exceptions above) no more eigenvalues/eigenvectors since the linear transformation restricts to a rotation on the remaining two dimensional subspace.

In $\Bbb R^n$ one (afaik) usually does not speak of rotations, because there are not really axes of rotation any longer.

Jonas Linssen
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  • "In $\Bbb R^2$ a rotation matrix does not have any eigenvectors/eigenvalues in $\Bbb R$" ... with some exceptions as well in higher dimensions. – Michael Hoppe Jun 13 '20 at 09:02
  • @MichaelHoppe right, I removed identity and reflection. What do you mean with higher dimension in $\Bbb R^2$? – Jonas Linssen Jun 13 '20 at 09:19