# Why PCA(Principal component analysis) can reduce the linear relationship between variables

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!