554 Chapter 19The matrix eigenvalue problem
EXAMPLE 19.18Show that the following matrix is Hermitian:
We have
0 Exercise 40
Unitary matrices
A complex square matrixUis called unitary when its Hermitian conjugate is equal to
its inverse,
U
†1 = 1 U
− 1(19.58)
The characteristic property of a unitary matrix is that its columns (and its rows)
form a unitary system of orthonormal vectors as defined by (19.56) (compare the
discussion of orthogonal matrices in Section 18.6). For order 3, let
(19.59)
wherea
†a 1 = 1 b
†b 1 = 1 c
†c 1 = 11 anda
†b 1 = 1 b
†c 1 = 1 c
†a 1 = 10. The Hermitian conjugate of Uis
(19.60)
and
(19.61)
0 Exercises 41, 42
UU
a
b
c
abc
aa ab ac
†††††††==()
bba bb bc
ca cb cc
††††††
=
100
010
001
=I
U
a
b
c
††††==**
aaa
bb
12312bb
ccc
3123
Uabc==
()
abc
abc
abc
1 11222333AA(A)
†=
−
,==
−
323
23 1
323
23 1
i
i
i
i
T
=A
A=
−
323
23 1
i
i