Can two different data sets get the same eigenvector in PCA? As we know, we can get the same eigenvector if we apply PCA to the same data. But, is it possible that we get the same eigenvectors after we apply PCA to two totally different data sets (still same dimension)?
 A: Any pure rotation of the dataset would also give the same set of principal components, in which case the answer would be trivially "yes", depending on the definition of "totally different" (arguably if two datasets have the same principal components then they can't be "totally" different).
A: My answer to this question provides one example.
Two data sets, both containing two variables, with the same sign correlation (positive or negative or zero), will get the same PCs if you base your PCA on the correlation matrix (i.e. if you standardise your variables).  The two PCs are always $\pm$45 degree rotations, with the first PC being the rotation with same sign as correlation, and second PC being the rotation with opposite sign.  The size of the correlation is informative about the strength of the decomposition (via % variance explained), but totally uninformative about the direction.
Another example, in any dimension is if the correlations are all equal.  The correlation matrix then has two distinct eigenvalues $\lambda_1=1+(d-1)\rho$ (multiplicity $1$) and $\lambda_2=(1-\rho)$ (multiplicity $d-1$), with normalised eigenvector $e_1= \frac{1}{\sqrt{d}}(1,\dots,1)^T$.  The remaining $e_2,\dots,e_d$ are not unique, but span the subspace orthogonal to $e_1$ (they are still pairwise orthogonal because the matrix is symmetric and real).  One way to do this is for $j=2,\dots,d$
$$e_{j1}=\frac{1}{\sqrt{2}},e_{jj}=-\frac{1}{\sqrt{2}},e_{jk}=0\forall k\notin\{1,j\}$$
You can then create an infinite number of solutions by arbitrarily rotating this one (in $d-1$ dimesnions).  $(\tilde{e}_2,\dots,\tilde{e}_d)=(e_2,\dots,e_d)R$.
A: An eigenvector is just a statistic. You can get copies of almost any useful statistic from a wide variety of different data, as long as they are similar in the relevant manner.
The eigenvector represents a particular linear relationship between the variables. As long as you use variables with the same units and what appears to be the same linear relationship, after normalization, you will get the same eigenvector.
