19.7 Exercises 555
A real unitary matrix is an orthogonal matrix. Hermitian and unitary matrices play
the same roles for complex matrices as do symmetric and orthogonal matrices for real
matrices, and they have the same important properties:
(i) The eigenvalues of a Hermitian matrix are real.
(ii) The eigenvectors of a Hermitian matrix of order nform (or can be chosen to
form) a unitary system of northonormal vectors.
(iii) A Hermitian matrix Ais reduced to diagonal form by means of the similarity
(unitary) transformation
D 1 = 1 U
†AU (19.62)
where Uis the unitary matrix whose columns are the eigenvectors of A, and D
is the real diagonal matrix whose diagonal elements are the eigenvalues of A.
19.7 Exercises
Section 19.1
Find the inverse of the matrix of the coefficients, and use it to solve the equations:
- 2 x 1 − 13 y 1 = 18 2.x 1 + 1 y 1 + 1 z 1 = 16 3.w 1 + 1 x 1 + 1 z 1 = 12
4 x 1 + 1 y 1 = 12 x 1 + 12 y 1 + 13 z 1 = 114 x 1 + 1 y 1 + 1 z 1 = 16
x 1 + 14 y 1 + 19 z 1 = 136 w 1 + 1 y 1 + 1 z 1 = 13
w 1 + 1 x 1 + 1 y 1 = 14
Section 19.2
Find the eigenvalues and eigenvectors of the following matrices:
Section 19.3
Normalize the eigenvectors obtained in
12.Exercise 5 13.Exercise 7 14.Exercise 8 15.Exercise 9
Show that the sets of eigenvectors of the symmetric matrices are orthogonal in
16.Exercise 5 17.Exercise 7 18.Exercise 9
19.Given the three vectors
use Schmidt orthogonalization to (i)find new vectorsx′
2
andx′
3
that are orthogonal tox
1
,
(ii)find the new vectorx
3
′′that is orthogonal to bothx
1
andx′
2
.
xxx
123
1
1
1
3
1
2
3
= 2
=
,,=
11
,
40 2
61 4
60 3
−−
−−
34
−− 45
030
303
030
120
210
021
31
13
31
− 13
20
03 −
22
13