Chapter 7. The Eigenvalue Problem

7. The Eigenvalue Problem, December 17, 2009 22
Exercise 8 Verify that (1) Ms(i) =λ i s(i), i =1, 2, 3, with λ 1 =3,λ 2 =0,
λ 3 =0;and(2)s(i) | s(j) = δ ij , i, j =1, 2, 3.
What you have seen in this example is general.
1. The eigenvectors of a Hermitian matrix are linearly independent.
2. The eigenvectors corresponding to different eigenvalues are automatically
orthogonal.
3. The degenerate eigenvectors are not necessarily orthogonal to each
other. One can always convert them (by Gram-Schmidt) into a set
of degenerate eigenvectors that are orthogonal to each other and to all
other eigenvectors.
4. Any nonzero vector can be normalized by dividing it by its norm
(x/ √ x · x is normalized).
§ 16 Find the eigenvalues of the matrix M (Eq. 87) “by hand”. You have
learned enough to be able to use the computer and generate orthonormal
eigenvectors. It would not hurt, however, to understand how degeneracy
appears, by doing an eigenvalue calculation, step by step, to look at the
details.
The eigenvalue problem for matrix M is
Mx = λx (112)
For the matrix given by Eq. 87, this is the same as the system of equations
(1 − λ)x 1 + x 2 + x 3 = 0 (113)
x 1 +(1− λ)x 2 + x 3 = 0 (114)
x 1 + x 2 +(1− λ)x 3 = 0 (115)
where x 1 , x 2 , x 3 are the components of the vector x.
The characteristic polynomial is
⎛
⎜
det[M − λI] =det⎝
1 − λ 1 1
1 1− λ 1
1 1 1− λ
⎞
⎟
⎠ =3λ 2 − λ 3 (116)