I admit this isn't too urgent, so apologies, but I'm posting here because I have exams quite soon and my inability to do this particular problem is getting a bit frustrating, not to mention worrying.
Anyway, I understand how to work out eigenvectors for 2x2 matrices well enough, I think, but have absolutely no idea how to go about it for 3x3 matrices. I've looked around online but I either don't understand the notation used or it's just not dumbed down enough for me, I suppose...
Specifically, anyway, the problem is as follows:
Given the 3x3 matrix:
( 2 -4 0 )
( -4 2 0 )
( 0 0 4 )
I have to work out the eigenvalues and corresponding eigenvectors. I've come up with the eigenvalues 4, 6 and -2, and used Mathematica to check these which are apparently right. But where do I go from here to get the eigenvectors?
I only need help with one of them if someone could just explain the method to me - in especially simple terms note, since I'm apparently incapable of understanding anything more complicated.
Starting with the eigenvalue lamda = 4, for example, and calculating (A - lamda I)X (where A is the original matrix, I is the identity) I've come up with these equations:
-2x_11 - 4x_21 = 0
-4x_11 - 2x_21 = 0
I'm using _ to indicate subscript by the way, not sure if there's a clearer way to do this or not. But anyway these equations seem to contradict each other, so I'm not even sure I've started out properly. I assume the bottom value of the resulting column vector would be 0, but that's as far as I can get.