If we are given a set of vectors $v_1$, $v_2$....$v_n$ how many matrices can we construct with the given set of vectors as the eigenvectors of the matrix?
If a set of eigenvalues and eigenvectors were given then we could have found a unique matrix. In this case no information on eigenvalues is given and hence they can be chosen arbitrarily. So, I think there are an infinite number of matrices which satisfies the required condition. Is my conclusion correct?