# Question on orthonormal basis of eigenvectors for a normal operator T

• Apr 17th 2013, 11:28 PM
Migno
Question on orthonormal basis of eigenvectors for a normal operator T
Hello,
any help on this question is greatly appreciated.
"For each linear operator T on an inner product space V, determine wheter T is normal, self-adjoint, or neither. If possible, produce an orthonormal basis of eigenvectors of T for V and list the corresponding eigenvalues.

(c) V=C2 and T is defined by T(a,b) = (2a + ib , a + 2b)"

So, I know T is not self-adjoint and that it is normal since TT*=T*T and since it's over C2, there has to be an orthonormal basis of eigenvectors, but I can't find them.
Using the standard basis B, A = [T]B = [2 i]
[1 2]

and det(A-tI) = t^2 - 4t + 4 - i
If I use the quadratic formula I get 2 +- (1+i)/sqrt(2)
Should I pursue this further or am I doing something wrong. I am unable to find the eigenvectors.
• Apr 18th 2013, 01:43 AM
chiro
Re: Question on orthonormal basis of eigenvectors for a normal operator T
Hey Migno.

Can you show us your row reduction operations for both eigen-vectors? (The approach you used to get the eigen-values is OK).
• Apr 19th 2013, 10:31 AM
Migno
Re: Question on orthonormal basis of eigenvectors for a normal operator T
I see, it just looked funny haha. I got it now, thanks.