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Are eigenvectors (principal components) of PCA orthonormal or only orthogonal ? Or only some of them are orthonormal or they are orthonormal if data were normalized before doing PCA ?

Daniel Wiczew
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    eigenvectors are by definition the orthonormal version of the basis set vector. It is not that they are guaranteed to be orthonormal, rather any non orthonormal vector cannot be an eigenvector (but may be a scaled eigenvector). – ReneBt Jul 02 '21 at 08:54
  • @ReneBT I don't see something. According to this post: https://math.stackexchange.com/questions/157382/are-the-eigenvectors-of-a-real-symmetric-matrix-always-an-orthonormal-basis-with eigenvectors, can be non-orthonormal also. – Daniel Wiczew Jul 02 '21 at 09:18
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    PCA separates magnitude (eigenvalues) from direction (eigenvectors). The direction (angle) cosines are the eigenvector values, and the length of each (eigen)vector is 1. So they are orthonormal "by definition". You can scale each eigenvector as you like subsequently. The loading-vector is the eigenvector scaled up to its corresponding eigenvalue: https://stats.stackexchange.com/q/143905/3277. – ttnphns Jul 02 '21 at 11:24
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    A careful reading of the post on [math.se] does not contradict the comments. The point is that the possibility of non-orthonormal eigenvectors arises only under "degeneracy," which is when one or more eigenspaces has a dimension exceeding 1. But in such cases PCA *always* selects orthogonormal bases anyway. – whuber Jul 02 '21 at 12:44

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