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PCA procedure includes SVD of Covariance matrix. Based on eigenvalues we can find a proportion of variance explained by related Principal Components (eigenvectors).

variance explained by PC1 (in case symmetric matrix) = eigenvalue1 / (sum of all eigenvalues) * 100%

We form a covariance matrix as [ [SD1^2, RO*SD1*SD2], [RO*SD1*SD2, SD2^2]] (SD = standard deviation, RO = Pearson correlation coefficient)

It is clear that if RO = 1 then PC1 explains ~100% of the variance.

Can you please suggest intuition and a general (mathematical) connection between Pearson correlation coefficient and proportion of variance explained in PCA?

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