rotation.

It will now be shown that a proper rotation matrix has at least one invariant vector , i.e., . Because this requires that , we see that the vector must be an eigenvector of the matrix with eigenvalue . Thus, this is equivalent to showing that .

Use the two relations
:for any 3 × 3 matrix A and
:(since ) to compute
:

This shows that is a root (solution) of the characteristic equation, that is,
:

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