Find a possible Jordan basis for the linear operator $T$ such that:
- $T(x, y, z, t) = (2y, −2x + 4y, z + t, z + t)$
Is there an specific method to find a Jordan basis? Since I'm teaching myself I'm not aware of any sort of procedure into finding one.
However I managed to calculate the characteristic polynomial of $T$, $p(t)= t(t-2)(t-2)(t-2)$ concluding that the proper sub-space associated to $t=0$ is generated by $[(0,0,1,-1)]$ and the one associated to $t=2$ is generated by $[(1,1,0,0),(0,0,1,1)]$ which is why $T$ is non-diagonizable.
That's as far as I could get by myself, so I would appreciate any kind of explanation on how to find a Jordan basis or simply solving this exercise. I'm not used to speaking English so feel free to edit the question as I know the idea can be transmitted in a much better way.