Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 26
remaining ones. The eigenvector given by them has an undetermined component
(since we lost an equation) and we find it by imposing normalization. If
an eigenvalue is doubly degenerate, we lose two equations. The eigenvectors
have two undetermined components and we determine them by forcing the
degenerate eigenvectors to be orthonormal. If you have a 9 × 9 matrix and
an eigenvalue is six-fold degenerate, you lose seven equations; and so forth.
Appendix 7.1. A few properties of determinants
1. The determinant of a Hermitian or a unitary matrix is equal to the
product of its eigenvalues. Therefore the determinant of a Hermitian
matrix is a real number (even when the matrix has complex elements)
and the absolute value of the determinant of a unitary matrix is equal
to 1.
2. We have
det A −1 = 1
det A
where by writing A −1 we are asserting that it exists.
(141)
3. The determinant of a product is the product of the determinants:
det(AB) =(detA)(det B) (142)
4.Ifarowofamatrixisalinearcombination of other rows, then the
determinant of the matrix is zero.
5. If a column of a matrix is a linear combination of other columns, then
the determinant of the matrix is zero.
Exercise 11 Prove Eq. 141 when A is the matrix representation of  =
α |α α α|.
Exercise 12 Show that det(A −1 BA) =detB.