# PCA performed on a configuration with scaled axes

Suppose a configuration $X\in\mathbb{R}^{n\times 2}$ is output of PCA on some high-dimensional data $Y\in\mathbb{R}^{n\times h}$. Note that this PCA is performed by $$X=Y\cdot U,$$ where columns of $U$ are eigenvectors of $Y^TY$ corresponding to its 2 dominant eigenvalues. Since $X$ is a result of PCA, then $X^TX=diag(a, b)$, for some $a, b\in\mathbb{R}$. If one now performs PCA on $X$, then the matrix $V$ containing eigenvectors of $X^TX$ would be columns of an identity matrix, and the result would be intact.

Suppose that a configuration $X$ is normalized (axes of unit length) by $\hat{X}=X\cdot diag^{-1/2}(a, b)$. If one then performs PCA on $\hat{X}$, then $\hat{X}^T\hat{X}$ would be an identity matrix. In other words, by normalizing columns of $X$ one cannot affect the PCA result except for the aspect ratio change (stretching of axes). If fact, any arbitrary stretching of axes of $X$ would have no effect since also then $\hat{X}^T\hat{X}$ would be a diagonal matrix. Am I missing something?

-