tickle_monste
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It was pretty cool to stumble upon Euler's formula as the eigenvalues of the rotation matrix.
det(Rot - kI) = (cos t - k)2 + sin2t
=k2-2(cos t)k + cos2t + sin2t
=k2-2(cos t)k + 1
k = {2cos t +/- \sqrt{4cos^2(t) - 4}}/2
k = cos t +/- \sqrt{cos^2(t) - 1}
k = cos t +/- \sqrt{cos^2(t) - cos^2t - sin^2(t)}
k = cos t +/- \sqrt{-sin^2(t)}
k = cos t +/- i sin t = e(+/-)it
I was wondering what the eigenvalues are for the rotation matrix in 3D, and if there's a 3D equivalent to Euler's formula.
det(Rot - kI) = (cos t - k)2 + sin2t
=k2-2(cos t)k + cos2t + sin2t
=k2-2(cos t)k + 1
k = {2cos t +/- \sqrt{4cos^2(t) - 4}}/2
k = cos t +/- \sqrt{cos^2(t) - 1}
k = cos t +/- \sqrt{cos^2(t) - cos^2t - sin^2(t)}
k = cos t +/- \sqrt{-sin^2(t)}
k = cos t +/- i sin t = e(+/-)it
I was wondering what the eigenvalues are for the rotation matrix in 3D, and if there's a 3D equivalent to Euler's formula.