Eigenvalue of 3D rotation matrix

• tickle_monste
In summary, the conversation discusses the eigenvalues of the rotation matrix and the possibility of a 3D equivalent to Euler's formula. It is noted that a rotation can be represented as complex multiplication and that a 3D rotation matrix can be expanded to a 2D rotation matrix. The conversation also mentions the real-valued eigenvalue and the potential extension of Euler's formula through quaternions.
tickle_monste
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.

Actually, I suppose it is not that strange when you look at a rotation as complex multiplication.
If you write a two-dimensional vector v = (x, y) as z = x + iy, then rotation over an angle t can be written either as R(t) v, where R is the 2d rotation matrix. But you can also write it as eit z.

A 3D rotation matrix, in the appropriate basis, looks like
$$R_3(t) = \begin{pmatrix} R_2(t) & 0 \\ 0 & 1 \end{pmatrix}$$
where R2(t) is the 2-dimensional rotation matrix and the z-axis is the rotation axis (i.e. rotation in the (x,y) plane). So, by expansion along the last row or column,
det(R3 - k I) = (1 - k) det(R2 - k I)
which you can work out in terms of your previous result. (And then add that an arbitrary change of basis does not alter the eigenvalues).

In addition to the two imaginary eigenvalues, the most intuitive one (for me) is the real valued one -- it's 1, with eigenvector along the axis of rotation. You can see that one just by noting that rotating won't change a vector along the axis of rotation.

edit:
tickle_monste said:
if there's a 3D equivalent to Euler's formula.
Quaternions could be considered an extension of Euler's formula ...

Last edited:

What is the eigenvalue of a 3D rotation matrix?

The eigenvalue of a 3D rotation matrix is a scalar value that represents the amount by which a vector is scaled when multiplied by the matrix. It can also be thought of as the amount of rotation around a particular axis.

How do you calculate the eigenvalues of a 3D rotation matrix?

The eigenvalues of a 3D rotation matrix can be calculated using various methods, such as the characteristic polynomial or the trace-determinant method. These methods involve finding the roots of the characteristic equation or solving a system of equations to determine the eigenvalues.

What do the eigenvalues of a 3D rotation matrix tell us?

The eigenvalues of a 3D rotation matrix provide important information about the rotation, such as the axis and angle of rotation. They also help determine the stability of the rotation and how it may affect other vectors in the space.

Can a 3D rotation matrix have complex eigenvalues?

Yes, a 3D rotation matrix can have complex eigenvalues, which represent rotations in higher dimensions. In this case, the eigenvalues will come in pairs of complex conjugates.

How do the eigenvalues of a 3D rotation matrix relate to its eigenvectors?

The eigenvalues of a 3D rotation matrix correspond to its eigenvectors, which are the directions along which the matrix only scales the vector but does not change its direction. The eigenvectors and eigenvalues together make up the eigendecomposition of the rotation matrix.

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