Suppose there is a transformation $T$ and let $A$ be a matrix representation of $T$ with chosen basis. If I find out the eigenvalues of matrix $A$, these eigenvalues will be the eigenvalues of the transformation $T$?
Then what about eigenvectors of $T$? As far as I know, similar matrices have same eigenvalues, so any matrix representation of $T$ with different basis has same eigenvalues, but eigenvectors corresponding to eigenvalues are dependent of matrix representation.
Then, what can I say about eigenvectors of $T$ by just looking at the eigenvectors of matrices?