I am looking at this wikipedia page
http://en.wikipedia.org/wiki/Matrix_decomposition#Eigendecomposition
Eigendecomposition
Also called spectral decomposition
Applicable to: square matrix A with distinct eigenvectors (not necessarily distinct eigenvalues).
What is the conditions of having distinct eigenvectors? Is it possible to have different eigenvectors while having the same eigenvalues?
It depends on the conditions you have available to you, that you have information about.
(Sufficent)You can compute the characteristic polynomial $Det ( A- \lambda I) $ and check there are different eigenvalues, i.e., no n-ple roots for $n \times n$ matrix.
You can also just compute the eigenspaces if you have a repeated root; the eigenspaces associated to different eigenvalues are linearly independent, i.e., the basis vectors are independent of each other.