U ¼
1
⋱
1
0
Á Á Á
1
1
⋮
⋱
⋮
1
1
Á Á Á
0
1
⋱
1
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
,
ð14:66Þ
where except (i, j) and ( j, i) elements equal to 1, all the off-diagonal elements are
zero. If operated from the left, U exchanges the i-th and j-th rows of the matrix. If
operated from the right, U exchanges the i-th and j-th columns of the matrix. Note
that U is at once unitary and Hermitian with eigenvalue 1 or À1. Note that U
2
¼ E.
This is because exchanging two columns (or two rows) two times produces identity
transformation. Thus performing such unitary similarity transformations appropriate
times, we get
~
D n
α 1
⋱
α 1
α 2
⋱
α 2
⋱
α s
⋱
α s
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
:
ð14:67Þ
The matrix is identical to that represented in (12.181).
In parallel, V
n is decomposed to mutually orthogonal subspaces associated with
different eigenvalues α 1 , α 2 , Á Á Á, and α s such that
V
n
¼ W α 1 È W α 2 È Á Á Á È W α s :
ð14:68Þ
This expression is formally identical to that represented in (12.191). Note,
however, that in (12.181) orthogonal subspaces are not implied. At the same time,
A n is reduced to
560
14 Hermitian Operators and Unitary Operators