Directly from the definition of "eigenvector": , if is an eigenvector of A corresponding to eigenvalue 2, then
so we have the three equations 2x+ 2y- 2z= 2x, y= 2y, z= 2z. The last two give y= 0, z= 0, of course, and 2x= 2x+ 0- 0 is satisfied for any x.
That is, of course, equivalent to saying
which gives the equations 2y- 2z= 0, y= 0, z= 0. That says nothing about x so x can be any number.