Hello!
"UNITARY OPERATOR IN QUANTUM MECHANICS"
We are in the third part of the problem
(c) Obtain an unitary transformation to diagonalize the "Z" matrix
"v_1" and "v_2" are the eigenvectors
We substitute the matrix columns by the eigenvectors
We obtain "U" and "U^+" its complex conjugate
Now we multiply both matrices to check if  "U" is unitary
We make rows by columns
We obtain the identity matrix
Now we pay our attention in the unitary transform
Three matrices are multiplied
We substitute
We may write a common factor
Rows by columns
Now the second row
We simplify
First row
Now the second product
It is diagonal
We can write the final result
We simplify again
The diagonal elements are the eigenvalues
