# Are the original features a linear combination of the principal components?

In PCA, we often think of the principal components / eigenvectors as a linear combination of the original features in vector space. Is the reverse true as well, i.e., is each feature a linear combination of the principal components?

• Do you mean exactly or approximately? – whuber Jun 29 '20 at 21:54
• @whuber Hmm, I originally meant theoretically, so exactly. But your question makes me think that the answer is "no" to exactly, but that it can be approximately written as a linear combination of the PCs, so I would be interested in that answer as well. – David Jun 29 '20 at 22:00
• The answer depends on how many PCs you retain. The whole idea behind PCA is to approximate the original features using a smaller number of PCs. By taking all the PCs, nothing is lost. Have you looked at our higher voted threads on PCA? – whuber Jun 29 '20 at 22:05
• @whuber Yup. It's actually this stats.stackexchange.com/questions/183236/… one that inspired my question. The answer by amoeba says "The intuition is that PCA seeks to represent all $n$ data vectors as linear combinations of a small number of eigenvectors, and does it to minimize the mean-squared reconstruction error." I've never seen it phrased this way before, but it seems to be saying the features are represented as a linear combination of the PCs. – David Jun 29 '20 at 22:14
• That's right. That's the basis of just about every plot of PCs that has ever been produced: the data are approximated as points in a space spanned by two (or sometimes three) PCs. – whuber Jun 29 '20 at 22:40