PCA instead of just one main shock?

[matdehaven]

After identifying the main shock, I simply construct a nullspace matrix for that first eigenvector. This assures that the new rotation matrix is orthonormal. However, it seems like I could just take all the Eigen vectors and make a matrix with them? I think this is still orthonormal, making it a valid rotation. This would have the added benefit that now the 2nd, 3rd, etc shocks can be interpreted as 2nd, 3rd, etc principal components. That's not very useful, but seems better than uninterpretable values.

[fernando-duarte]

If the matrix you are taking the eigenvalues of is real symmetric with all eigenvalues different, then the eigenvectors are always orthogonal and we get the nice PCA interpretation automatically. With repeated eigenvalues, it is always possible to pick some eigenvalues that are orthogonal with a little bit more work (and I presume some standard R command can choose a set of orthogonal eigenvalues for you in this case, or perturb a bit to numerically get rid of this issue).

I think we should not normalize the length of the eigenvectors (i.e. orthogonal but not necessarily orthonormal eigenvectors).