# Can the dimensions be reduced with PCA without loss of data if $N>p$? [closed]

Say I have a matrix of feature vectors where I have 1000 samples (feature vectors) while dimension of each feature vector is 10. Can I apply PCA to reduce the dimensions further?

## closed as unclear what you're asking by amoeba, John, kjetil b halvorsen, Peter Flom♦Feb 6 '17 at 13:32

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• What do you mean "without loss of data"? Can you reduce the dimensionality with PCA? Yes you can. Will you lose some information? Yes you will. – amoeba Feb 5 '17 at 23:38
• @amoeba what if the data lie on a linear sub space? – Sycorax Feb 5 '17 at 23:40

Consider looking at the eigenvalues of $(X^TX)$. If the eigenvalues drop off sharply then you can possibly drop your dimension further while retaining total variance. If the eigenvalues are roughly equal then PCA won't help you at all.