Quote by HallsofIvy
No. Two matrices are similar if and only if they have the same eigenvalues and corresponding eigenvectors.

Would you mind clarifying this point? It's well known that a similarity transformation preserves the spectrum, but the eigenvectors?
The matrices
[ 0 1 ]
[ 0 0 ]
and
[ 0 0 ]
[ 1 0 ]
are similar via the permutation matrix
[ 0 1 ]
[ 1 0 ],
but they don't share the same eigenvectors.