For a given centered configuration of points $X\in\mathbb{R}^{n\times 3}$, the covariance matrix is denoted by $S=\frac{1}{n}X^TX$. Recall that the 2D PCA solution is obtained by $Y=X\cdot U$, where $U\in\mathbb{R}^{3\times 2}$ is a matrix of orthonormal eigenvectors of $S$ associated with its 2 dominant eigenvalues $\lambda_1, \lambda_2$.
Now, suppose that $Y_1\in\mathbb{R}^{n\times 2}$ is obtained by normalizing each axis of $Y$. The normalization factor would be $1/\sqrt{\lambda_i}$, $i\in\{1,2\}$ (please correct me if I'm wrong here).
Also, suppose that $Z$ is obtained by normalizing each axis of $X$ to unit length. A 2D PCA solution on $Z$ is denoted by $Y_2$. Are $Y_1$ and $Y_2$ the same? I obtain that the results are different, but it hold that the axes of both $Y_1$ and $Y_2$ are orthonormal. If so, what is the distinguishing characteristic of $Y_2$ relative to $Y_1$?
