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Assuming we have centralized data,the covariance matrix of the sample is X'X. This is Because: $$ Cov(X)=\frac{1}{n-1} \begin{pmatrix} X_1'X_1 &...&X_1'X_n \\ ...&...&...\\ X_n'X_1&...&X_n'X_n \end{pmatrix} $$ In PCA,we make an eigendecomposition $X'X=\phi \Lambda \phi'$ to turn the covariance matrix of the sample into diagonal matrix $\Lambda$ Bacause the Non-diagonal element of the covariance matrix of the sample is zero,we reduce the linear relationship between new variables (we set $Z$) without deleting some variebles

But I dont think so. If the $\Lambda$ has a very small eigenvalue $\lambda_i$,we have $Z'Z\phi_i=\Lambda\phi\approx0$

$\Rightarrow Z\phi\approx0 \Rightarrow$ The new variable is approximately linearly dependent

I don't know what's wrong. I hope you can help me.Thank you!

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