E x E y
À
Á
1
À cos δ
À cos δ
1
E x
E y
¼ E
2
1 sin
2
δ:
ð7:79Þ
We obtain eigenvalues λ of the matrix of (7.79) such that
λ ¼ 1Æ j cos δ j :
ð7:80Þ
Setting À
π
2
δ
π
2 , we have
λ ¼ 1 Æ cos δ:
ð7:81Þ
The corresponding normalized eigenvectors v 1 and v 2 (as a column vector) are
v 1 ¼
1
ffiffi ffi
2
p
À
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A and v 2 ¼
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A :
ð7:82Þ
Thus, we have a diagonalizing unitary matrix P such that
P ¼
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
À
1
ffiffi ffi
2
p
1
ffiffi ffi
2
p
0
B
B
@
1
C
C
A :
ð7:83Þ
Defining the above matrix appearing in (7.79) as A such that
A ¼
1
À cos δ
À cos δ
1
,
ð7:84Þ
we obtain
P
À1 AP ¼
1 þ cos δ
0
0
1À cos δ
:
ð7:85Þ
Notice that eigenvalues (1 + cos δ) and (1 À cos δ) are both positive as expected.
Rewriting (7.79), we have
E x E y
À
Á
PP
À1
1
À cos δ
À cos δ
1
PP
À1
E x
E y
290
7 Maxwell’s Equations