Answer: Anything except zero. When finding eigenvectors, you should expect to find infinitely many solutions. After all, your eigenvalues are chosen so that the system you're solving, when trying to find eigenvectors, has infinitely many solutions! The reason you can't allow zero in this particular case is that if all three components are zero, you've got yourself a zero vector, which by definition is not an eigenvector.So U1 = 0 and U3 = 0, but what does U2 equal?

For this eigenvalue of 2, this is not an eigenvector. It might be an eigenvector corresponding to 3, though. If you look at your system ofU1 = -1 U2 = 2 and U3 = 3.

2U1 + U3 = 0

-2U1 - U2 = 0

-2U1 -U3 =0,

the first and third equation are the same. So you get

2U1 + U3 = 0

-2U1 - U2 = 0

Multiply the second row by -1:

2U1 + U3 = 0

2U1 + U2 = 0

From this, we can tell that U3 = U2, since they both equal -2U1. Therefore, the answer U1 = -1 U2 = 2 and U3 = 3 is impossible for this eigenvalue.