Question 1: (Source: How to calculate eigenvectors for equal eigenvalues?)
In the matrix posted there, I am wondering if the solution (which was accepted too...) posted has a mistake.
If we have $[1,\frac{1}{2},0 \ | \ 0]$ as our upper row, then $x + \frac{1}{2} y = 0$, so $2x + y = 0$, so $y = -2x$. Shouldn't this make the basis for the eigenspace as $[-2,1,0]$ then, and not $[1,2,0]$? Or have I made a mistake?
Question 2: (Source: How can I find the eigenvectors of this Matrix?)
I read the solutions/comments, and they all seem to point out how OP has made an error in calculating the dimension, but never seem to answer his question about how to get the eigenspaces when you have $[0,0,1 \ | \ 0]$ or the like... In this case, would it just mean $x_3=0$? I too am confused as to how Matlab obtained those eigenvectors - moreover, if the dimension is $1$, how did it come up with two of them? Any help would be appreciated.