What is the difference between eigenvectors and principal component. I got confused about this point because some researches reported that the principal components are the same eigenvectors of covariance matrix whereas, the others say that the principal components are the yield of eigenvectors (weights) of covariance matrix multiplied with the original data.

Please if some one can help me in this point.

  • $\begingroup$ when i learnt it, it was just the eigenvectors of the matrix. but the matrix does not have to be covariance matrix. it can be correlation matrix, because covariance matrix depend on the scale of the variable. (but use covariance if all the variables are of the same dimension and have the same unit) $\endgroup$ – Lost1 Apr 10 '13 at 12:42
  • $\begingroup$ There is also a (somehow sophisticated) difference between "principal axis" and "principal component". While the view at the statistical "items" leads to the notion of "components" (and then to the "principal components") the view into an (euclidean) coordinates-system in which the items are located leads to the notion of "axes" (of that coordinate system) and the "principal axes". The model of "eigenvectors" is here identified with that view at axes of an euclidean system/vector space, and the "principal axes" identify then with the eigenvectors. (...) $\endgroup$ – Gottfried Helms Apr 10 '13 at 14:03
  • $\begingroup$ (...) Then a "principal component" is (according to the "item-view") one (latent, not directly measured) item (or "component"), whose only nonzero coordinate is that on the principal axis. The latter identity may be the reason for the (irritatingly interchangeably) common usage of "principal component" and "eigenvector". Additionally, the "principal components" coordinate is then scaled by the eigenvalue associated with that specific eigenvector/on that n'th principal axis. $\endgroup$ – Gottfried Helms Apr 10 '13 at 14:07

Loosely speaking the eigenvectors are just the linear combinations of the original variables. Their eigenvalues which are associated with each principal component tell you how much variation in the data set is explained.

Please have a look at the very good answers here.


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