Exercises involving eigenvectors

In the following exercises 'verify' means using a hand calculator. They are intended to confirm understanding of the theory involved. Work though one or more examples to confirm the validity of the statement that follows.
1. Verify that the eigenvectors are normalized.
2. Verify that the eigenvectors are orthogonal to each other.
3. Verify that the eigenvalues are correct.
4. Verify that the eigenvectors diagonalize the Fock matrix.
5. Verify that the diagonal sum rule is obeyed; i.e., that the sum of the eigenvalues is equal to the sum of the diagonal matrix elements (the trace) of the Fock matrix.