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I am reading a chapter about principal component analysis (PCA). It states that for any random varible $X \in \mathbb{R}^p$ with $n$ observations, $E[X] = \mu$ and $Cov[X] = \Sigma$ the i-th PC is $y_i = \gamma_i^{T} X$, where $\gamma_i$ is the eigenvector corresponding to the i-th largest eigenvalue of the covariance matrix of $X$. Then, it is stated that the PC-transformation is defined as $Y = \Gamma^T (X - \mu)$.

This confuses me because in earlier studies I learned PCA is applied to centered data. So I always thought that the i-th PC corresponds to $y_i = \gamma_i^{T} Z$, where $Z$ is centered data.

So, I think the statement with the PC-transformation is right in the book but the first statement of which defines the PC is wrong and even a contradiction to the statement of the PC-transformation.

Could please somebody make clear whether I am right or wrong.

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    $\begingroup$ Don't feel confused. Centering or z-standardization or other possible transformation is itself not a part of PCA. It is the preprocessing which affects the results and their interpretation strongly, but still it is not an inherent part. The PCA in its clean view is just svd or eigen-decomposition. PCs obtained under different transformations of the data are different PCs. In both your formulas you could mean X as raw data or as X-mu, centered data. Both are correct PCA but are different analyses with different results. $\endgroup$ – ttnphns Jul 19 '15 at 12:22
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    $\begingroup$ Specifically of your 1st paragraph. The eigenvalues and eigenvectors of the covariance matrix are those of the data (X-mu)/sqrt(n) [or n-1 here, since we usually use this correction to compute covariance]. While eigenvalues and eigenvectors of the X'X/n matrix ("mean SSCP" = MSCP matrix) correspond to those of the data X/sqrt(n). $\endgroup$ – ttnphns Jul 19 '15 at 12:34
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    $\begingroup$ Note that division of data by sqrt(n) in my comment affect (scale down) eigevalues, not eigenvectors. $\endgroup$ – ttnphns Jul 19 '15 at 12:49
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    $\begingroup$ I think you are right, your book is inconsistent. What they call "PC" is not the result of what they call "PC-transformation". Your understanding is correct. $\endgroup$ – amoeba Jul 19 '15 at 13:13

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