This question is a little theoretical. Say I have 3 points that are bound within a certain 3D volume of some shape (i.e. box, cylinder, cone, etc). By using PCA, I find the most two dominant directions and reduce the data dimension to 2D.
All of this is done by the following:
- Compute 3x3 covariance matrix of the 3D data.
- Take eigenvectors and eigenvalues of the covariance matrix.
- Sort by eigs by descending order of eigenvalues.
- Take first two largest eigenvectors and multiply them with the data. The data is now 2 dimensions.
My question is: What properties are maintained through the projection? For example, if the original 3D data was bound by a radius of size r (say in the X and Y directions only), does this apply also to the 2D data? What happens to the range of the data? The distances? What can I infer about the new 2D data from knowing the characteristics of the 3D data?