19.3 Properties of the eigenvectors 537
(ii) The characteristic equation of the symmetric matrix
with nondegenerate eigenvalueλ
11 = 14 and the pair of degenerate valuesλ
21 = 1 λ
31 = 11.
The secular equations are
Forλ 1 = 14 : The solution isx 1 = 1 y 1 = 1 zand the eigenvector is , c
1arbitrary.
For λ 1 = 11 : Each secular equation givesx 1 + 1 y 1 + 1 z 1 = 10 , and every independent pair
of vectors that satisfy this condition is a solution for the degenerate eigenvalue;
for example,
and
0 Exercises 10, 11
Two types of matrices whose eigenvalues are of greatest interest in the physical
sciences are real symmetric matrices (symmetric matrices whose elements are all real
numbers) as in Example 19.4(ii), and the complex Hermitian matrices described in
Section 19.6. These matrices have real eigenvalues.
19.3 Properties of the eigenvectors
Property 1.If xis an eigenvector corresponding to eigenvalue λ, then kxis also
an eigenvector corresponding to the same eigenvalue, for any nonzero value of the
number k:
if Ax 1 = 1 λx then A(kx) 1 = 1 k(Ax) 1 = 1 k(λx) 1 = 1 λ(kx) (19.12)
Eigenvectors that differ only in a constant factor are not treated as distinct. It is
convenient and conventional to choose the factor kto make the eigenvector a unit
vector; that is, to normalize the vector.
x
331
2
3
=
−
c
x221
1
2
=
−
c
x
111
1
1
=
c
()
()
()
2
2
2
0
0
0
−+
−
+−
=
=
=
λ
λ
λ
x
x
x
y
y
y
z
z
z
211
121
112
211
12 1
112
−
−
−
is =
λ
λ
λ
(()()41 0
2−−=λλ