- #1
Samuel Williams
- 20
- 3
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
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