We have a matrix $A$ as below
$$\begin{bmatrix}1&2&2\\0&1&2\\0&0&2\end{bmatrix}.$$
I know the eigenvalues are 1 and 2 because matrix $A$ is a triangular matrix.
And by working it manually, my eigenvector for eigenvalue of $1$ is $(1,0,0)^T$, and eigenvector for eigenvalue of 2 is $(6,2,1)^T$.
However, when I used MATLAB to find my eigenvectors using >>[P D] = eig(A), I get {(1,0,0), (-1,0,0)} for eigenvalue of 1, and even more weird {(640/683, 640/2049, 320/2049)} eigenvector for eigenvalue of 2.
I know they're the same, since the eigenvectors in MATLAB are scalars of the eigenvectors above. However, my question is, why is there a difference between the MATLAB's eigenvectors and my own eigenvectors?