Specific case of Jordan normal form
I'm having trouble understanding Jordan normal form for matrices. I can prove it works for 3x3, but the general proof is a bit over my head, so I figured I'd try the 4x4 case. But even the really easy case, where matrix A has a single eigenvalue x and eigenspace of dimension 1, I can't prove it.
Obviously the Jordan normal form should look like:
x 1 0 0
0 x 1 0
0 0 x 1
0 0 0 x
but I'm not sure how to prove an arbitrary 4x4 matrix meeting those criteria can be put in this form.