Check the step again, where you find .
You are supposed to get that is 9 or 11 (10 is not an eigenvalue).
Suppose I had a linear map such that and I want to find the eigenvalues and eigenvectors of T.
I'm doing this without determinants. I have it that
After rearranging terms and doing simple elimination I have
I believe the eigenvalue is then . Next to find the eigenvectors I would find
To find I would end up having that
and so I have . I believe this means that all eigenvectors would be in the form (i.e. (1,1),(2,2)...)
Was everything I did correct?
Thank you.
No, you don't. Multiply the second equation by and the subtract it from the first equation. I suspect you did that but forgot the x in the second equation. You get . In order that that equation not have x= 0 as its only solution we must have
I believe the eigenvalue is then . Next to find the eigenvectors I would find
To find I would end up having that
and so I have . I believe this means that all eigenvectors would be in the form (i.e. (1,1),(2,2)...)
Was everything I did correct?
Thank you.
Thanks. I see what I did wrong, I did forget about the x.
For the eigenvectors for I would have to find which would be .
Setting (x+y,y+x)=(0,0) I would have x+y=0 and y+x=0 and so x=-y and y=-x. I believe the eigenvectors are then in the form (x,-x) i.e. (1,-1),(2,-2),...
Is how I found the eigenvectors correct?