Let $X=(x_1,x_2,...,x_n)$ where $x_i \in \mathbb{R}^d$ with $d>=2$. And where each $x_i = \sum_{j=0}^{d}a_j\cdot e_j$ where $(e_1,...,e_d)$ forms a basis in $\mathbb{R}^d$.
We define the first component of the PCA as the $d$-dimensional vector that maximizes the variance of the data projected into it.
Now suppose we do a change of basis $(e_1,...,e_d) \rightarrow (e^{*}_1,...,e^{*}_d)$. Is first (and all subsequent) components the same up to that change of basis?
ie: if we change basis, do we get the same PCA up to that change in basis?