e
G ¼
2 þ
ffiffi ffi
2
p
0
0
2À
ffiffi ffi
2
p
!
¼ 2 þ
ffiffi ffi
2
p
1 0
0 0
þ 2 À
ffiffi ffi
2
p
0 0
0 1
:
By a back calculation of U e
GU
{
¼ G, we get
G ¼ 2 þ
ffiffi ffi
2
p
1
2
ffiffi ffi
2
p
1 þ i
ð
Þ
4
ffiffi ffi
2
p
1 À i
ð
Þ
4
1
2
0
B
B
@
1
C
C
A þ 2 À
ffiffi ffi
2
p
Â
1
2
À
ffiffi ffi
2
p
1 þ i
ð
Þ
4
À
ffiffi ffi
2
p
1 À i
ð
Þ
4
1
2
0
B
B
@
1
C
C
A :
ð14:77Þ
Putting eigenvalues α 1 ¼ 2 þ
ffiffi ffi
2
p
and α 2 ¼ 2 À
ffiffi ffi
2
p
along with
A 1 ¼
1
2
ffiffi ffi
2
p
1 þ i
ð
Þ
4
ffiffi ffi
2
p
1 À i
ð
Þ
4
1
2
0
B
B
@
1
C
C
A , A 2 ¼
1
2
À
ffiffi ffi
2
p
1 þ i
ð
Þ
4
À
ffiffi ffi
2
p
1 À i
ð
Þ
4
1
2
0
B
B
@
1
C
C
A , ð14:78Þ
we get
G ¼ α 1 A 1 þ α 2 A 2 :
ð14:79Þ
In the above, A 1 and A 2 are projection operators. In fact, as anticipated we have
A 1
2
¼ A 1 , A 2
2
¼ A 2 , A 1 A 2 ¼ A 2 A 1 ¼ 0, A 1 þ A 2 ¼ E:
ð14:80Þ
Moreover, (14.78) obviously shows that both A 1 and A 2 are Hermitian. Thus,
Eqs. (14.77) and (14.79) are an example of the spectral decomposition. The decomposition is unique.
The Example 14.1 can be dealt with in parallel to Example 12.5. In Example 12.5,
however, an inner product space is not implied, and so we used an idempotent matrix
instead of a projection operator. Note that as can be seen in Example 12.5 that
idempotent matrix was not Hermitian.
14.4 Hermitian Matrices and Unitary Matrices
Of normal matrices, Hermitian matrices and unitary matrices play a crucial role both
in fundamental and applied science. Let us think of several topics and examples.
564
14 Hermitian Operators and Unitary Operators