I have been looking without success for an intuitive / geometric construction of the rotation axis of a given 3D rotation matrix.
To put the problem in more familiar terms, let's assume you have the canonical 3D orthonormal frame of reference (i,j,k) and another 3D orthornormal frame of reference (x,y,z). Then, the Matrix M whose rows are respectively the vectors x y and z is a rotation matrix.
I do know how to calculate the rotation axis of that matrix: it's the eigenvector for the eigenvalue 1 and can be found via solving the 3x3 linear system M.v = v for v.
However, solving that system gives me precious little geometric intuition about where that axis is and why. I would love to pick you guys collective brains to see if there is an intuitive, hopefully simple, construction of the rotation axis from the 6 vectors (i,j,k) and (x,y,z).
We're talking about 3x3 rotations in R3 vector space here, not affine matrices.