Eigenvalue and eigenvectors, bra-ket

[Samuel Williams]

Question

Consider the matrix $$
\left[
\matrix
{
0&0&-1+i \\
0&3&0 \\
-1-i&0&0
}
\right]
$$

(a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1>, |a2>, |a3>. Any degenerate eigenvalues?

(b) Show that the eigenvectors |a1>, |a2>, |a3> form an orthonormal and complete basis ;
|a1><a1|+|a2><a2|+|a3><a3|= I, where I is the 3x3 unit matrix,
and that <aj|ak> is the Kronecker delta function

(c) Find the matrix corresponding to the operator obtained from the ket-bra product of the
first eigenvector P=|a1><a1|. Is P a projection operator?

My attempt

I have done part (a). I got the eigenvalues as 3,√2,√2 with corresponding eigenvectors

(0 1 0) , ( (1-i)/√2 0 1 ) , ( -(1-i)/√2 0 1 )

Even after normalizing the vectors, I still can't work out part (b). I just don't get the 3x3 unit matrix.
Any help would be greatly appreciated

 

[suremarc]

The latter two eigenvectors aren't normalized!

Also, the eigenvalue corresponding to $\left(\begin{smallmatrix}\frac{1-i}{\sqrt{2}} & 0 & 1\end{smallmatrix}\right)$should have the opposite sign.

 

[Samuel Williams]

The eigenvalue should have a -, must have missed it.
I already normalized the vectors, giving

1/√(1-i)*((1−i√2) 0 1))

And it still doesn't seem to work out for me

 

[suremarc]

I already normalized the vectors, giving

1/√(1-i)*((1−i√2) 0 1))

I skimmed over that, my bad. Even so, it still isn't normalized--the magnitude is $2$, not $1-i$.
Use the equation $\|x\|=\sqrt{\langle x\;|\ x\rangle}$ to recover the norm on a Hilbert space. It should always be real-valued and nonnegative.

 

[Samuel Williams]

I managed to figure out where I have been going wrong thanks to you. I have been using Euclidean inner products instead of Hermitian inner products. Thanks for the help