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The thing I use to visualize an eigenvector is exactly that. A rotating planet. The eigenvector of the rotation matrix being the axis of rotation. It gets weir
by skywal_l 8y ago
The thing I use to visualize an eigenvector is exactly that. A rotating planet. The eigenvector of the rotation matrix being the axis of rotation.
It gets weird when thinking of 2D rotations though... Too complex for me!
- man-and-laptop 8y agoThe eigenvalues of a 2d rotation matrix are complex numbers. There's no point in trying to visualise the corresponding eigenvectors. You'll need 4 dimensions for that. (The 4 dimensions come from the fact that the eigenvectors of a 2d rotation matrix are elements of ℂ^2, which is topologically equivalent to (ℝ^2)^2, which is equiv to ℝ^4). Personally, my "intuition" is based on analogies with non-complex eigenvectors, and experience solving eigenvector problems algebraically without using pictures. Also, your 3d rotation example actually has three eigenvectors, two of which are complex. You've only found the one that's real.
- dhruvp 8y agoOP here: Your approach is pretty on point because Planetary Motion was actually the first recorded motivation for eigenvectors. Cauchy published a paper on planetary motion in 1829 which made use of what we now call Eigenvectors. See here for more info: https://www.maa.org/press/periodicals/convergence/math-origins-eigenvectors-and-eigenvalues https://www.maa.org/press/periodicals/convergence/math-origi...