There is a homework question for a course I am self studying (not a student) that is:
let our $n \times d$-dimensional data vectors be denoted by $x_1,\ldots,x_n$ and let $R$ be a $d \times d$ rotation matrix. For simplicity, you may assume that the $x_t$'s have been centered at $0$. Let $$x_t' = Rx_t + v$$ where $v$ is some fixed translation. Forming a second dataset. Now, for any $K$ we pick, let us use PCA on each of the two data sets to obtain $K$-dimensional projections $y_1,\ldots ,y_n$ and $y_1',\ldots,y_n'$, respectively.
Write down a relationship between the two PCA projection matrices $W$ and $W'$ in terms of the rotation matrix $R$ and the translation vector $v$. Explain mathematically how you arrived at this answer.
My answer is basically that while for the untransformed dataset we have that $W$ is the top $k$ eigenvectors of $S[\mu] (S[\mu]$ is the covariance matrix of the untransformed dataset centered at $\mu (\mu$ is $0$ for untransformed dataset)) that $W'$ will be the top $k$ eigenvectors of the matrix $(R^T) \times S[R^T(\mu-v)] R$. ($R^T$ is $R$ transposed). I did this by applying the transformation in the definition of the covariance matrix $S'$ to find a relation between $S'$ and $S$.
The goal of the problem is to show the rotational and translational invariance of PCA. Can anyone give an explanation for this?