Suppose a matrix $X\in\mathbb{R}^{n\times 3}$ is given as a Principal Component Analysis (PCA) projection from some high dimensional space. The 2D PCA solution on X, say $Y\in\mathbb{R}^{n\times 2}$ would simply correspond to the first two columns of $X$.Now, suppose the configuration is shifted such that the origin corresponds to an arbitrary point. I want to mathematically state (via PCA) that by changing the origin of $X$ (3D data), the new 2D PCA projection $Y'\in\mathbb{R}^{n\times 2}$ simply corresponds to two first columns on $X$ subject to rigid transformation (ie. rotation, reflection, shifting). The reason for the perhaps unnecessary complication is the PCA assumption on the configuration centered at the origin. (In other words, I'm not sure if by getting rid of it, one might loosen the connection with PCA)To remind you, PCA would be obtained as $$Y=XU_S,$$where $U_S\in\mathbb{R}^{3\times 2}$ would contain eigenvectors of the correlation matrix $S=\frac{1}{n}X^TX$, where $X$ is supposed to be centered at the origin. The origin change would correspond to $X'=PX$, where $P=(I-1_np^T)$ denotes the projector, and $1_n=[1, \dots, 1]\in\mathbb{R}^n$, and $p^T1_n=1$. The new correlation matrix $$S'=\frac{1}{n}((PX)^T(PX))=\frac{1}{n}X^TPX=\frac{1}{n}X^T(J-1_np_1^T)X,$$where $P$ might be expressed as $P=J-1_np_1^T$, where $J$ is a projector with $p=\frac{1}{n}1_n$, ie. $J=(I-\frac{1}{n}1_n1_n^T)$. So, I would like to state that $Y'=(PX)U_{S'}$ corresponds to a different 2D viewpoint on the primary $X$. The difficulty, in my interpretation, lies in the effects of spectral decomposition of $S'$, and its possible effects on rigid transformation on primary $X$. Again, I apologize for the perhaps unnecessary complication.
Proof continuation for rigid transformation on PCA solution
linear algebra;matrices;eigensystem;high dimensional
In general, the principal components of $X$ and $X'$ with rows shifted by a fixed vector will not have any simple relationship.In applications it may therefore be meaningful (or even important) to shift all rows of $X$ by subtraction their mean before doing the prinicpal component analysis. This makes the result independent of the placement of the origin.It may even be useful (or necessary) to scale the columns after shifting to have norm 1. This makes the result also independent of the scaling of the variables.