# Are eigenvectors of PCA guaranteed to be orthonormal?

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 ?

• 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). Commented Jul 2, 2021 at 8:54
• @ReneBT I don't see something. According to this post: math.stackexchange.com/questions/157382/… eigenvectors, can be non-orthonormal also. Commented Jul 2, 2021 at 9:18
• 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: stats.stackexchange.com/q/143905/3277. Commented Jul 2, 2021 at 11:24
• A careful reading of the post on Mathematics 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
Commented Jul 2, 2021 at 12:44