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?