Matrix Proof
A spatial rotation is a linear map in one-to-one correspondence with a 3×3 rotation matrix R that transforms a coordinate vector x into X, that is Rx = X. Therefore, another version of Euler's theorem is that for every rotation R, there is a vector n for which Rn = n. The line μn is the rotation axis of R.
A rotation matrix has the fundamental property that its inverse is its transpose, that is
where I is the 3×3 identity matrix and superscript T indicates the transposed matrix.
Compute the determinant of this relation to find that a rotation matrix has determinant ±1. In particular,
A rotation matrix with determinant +1 is a proper rotation, and one with a negative determinant −1 is an improper rotation, that is a reflection combined with a proper rotation.
It will now be shown that a rotation matrix R has at least one invariant vector n, i.e., R n = n. Because this requires that (R − I)n = 0, we see that the vector n must be an eigenvector of the matrix R with eigenvalue λ = 1. Thus, this is equivalent to showing that det(R − I) = 0.
Use the two relations:
to compute
This shows that λ = 1 is a root (solution) of the secular equation, that is,
In other words, the matrix R − I is singular and has a non-zero kernel, that is, there is at least one non-zero vector, say n, for which
The line μn for real μ is invariant under R, i.e., μn is a rotation axis. This proves Euler's theorem.
Read more about this topic: Euler's Rotation Theorem
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