19.6 Complex matrices 553
The inner product is complex in general, but the quantity
(19.55)
is a positive real number, and defines the length, or norm, of the vector,
EXAMPLE 19.17For the vectorsa 1 = 1 (1 1 + 1 i, 3, 2i)andb 1 = 1 (2 1 − 1 i, 4i, −1),
0 Exercise 39
A vector divided by its norm is a unit vector, and two vectors are orthogonal when
their inner product is zero. The complex analogue of an orthonormal system of
vectors is then
(19.56)
Such a system of vectors is called unitary.
Hermitian matrices
A complex square matrix that is equal to its Hermitian conjugate is called a Hermitian
matrix,
A
†1 = 1 A (Hermitian matrix) (19.57)
A real Hermitian matrix is a symmetric matrix.
aa
klkl
kl
†=
=
≠
1
0
if
if
ab
†=− − ()( )
()−
−
132 =− −
2
4
1
ii 12
i
iii+++−−=+ 34 ii2 1 1 11()( )()i
bb
†=+ − − ()(
()−
−
241 =+ −
2
4
1
ii 22
i
iii))( )()+−44 1516122ii+ = + + =
aa
†=− − ()()
()
132 =− ++
1
3
2
ii 11
i
i
ii 332229415
2+−()()ii=++=
||a = aa
†.
=++aaa
122232aa
·==
aa
†()aaa
a
a
a
123123*** ==++aa aa aa
11 22 33