I am wondering if there is any way to mathematically express the change in direction of the principal components from the $2\times2$ covariance matrix to the correlation matrix. In other words, if $$\Sigma=(X-\bar X)^T(X-\bar X)/(n-1)$$ $$=~ \left( \begin{array}{ccc} \sigma^2_{1} & \sigma_{12} \\ \sigma_{12} & \sigma^2_{2} \end{array} \right) $$ and $$\lambda,e$$ are the eigen vector pairs of $\Sigma$ then $$Y_{\Sigma}= e^TX$$ are the principal components. Can these be re-expressed for the correlation matrix $$\rho=~ \left( \begin{array}{ccc} 1 & \rho_{12} \\ \rho_{12} & 1\end{array} \right) $$ in terms of the covariance matrix in this manner: $$Y_{\rho}= f(Y_{\Sigma})$$


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