Transcription of Eigenvalues and eigenvectors of rotation matrices
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Eigenvalues and eigenvectors of rotation matrices
Physics 116 AFall 2019 Eigenvalues and eigenvectors of rotation matricesThese notes are a supplement to a previous class handout entitled, rotation Matricesin two, three and many dimensions. In these notes, we shall focus on the Eigenvalues andeigenvectors of proper and improper rotation matrices in two and three The Eigenvalues and eigenvectors of proper and improper rotation matricesin two dimensionsIn the previous class handout cited above, we showed that the most general properrotation matrix in two-dimensions is of the form,R( ) = cos sin sin cos ,where 0 <2 ,(1)which represents a proper counterclockwise rotation by an angle in thex the eigenvalue problem,R( )~v= ~v.
tation that led to eq. (22). Moreover, the other two eigenvalues are complex conjugates of each other, whose real part is equal to cosθ, which uniquely fixes the rotation angle in the convention where 0 ≤ θ ≤ π. Case 1 corresponds to the identity (i.e. no rotation) and Case 2 corresponds to a 180 rotation about the axis nˆ. In Case 2 ...
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