Changing the basis in diagonalization: why doesn't it work?
Hint:
The matrix $A$ represents the linear transformation $L$ (this is not a matrix) in the standard basis, and in this standard basis the linear transformation $L$ is characterized by the eigenvectors and the eigenvalues of the matrix $A$. But, if you change the basis to a new basis $E$, te same linear transformation is represented by another matrix $B=EAE^{-1}$ that have different eigenvectors (that represent the same eigenspace but in the new basis), and in this new representation of the same eigenspace, the eigenvalues are, in general, different.