Page 485 - Electromagnetics

reader can verify that any dyadic can be decomposed into symmetric and antisymmetric
parts as

1 T 1 T
¯ a = ¯ a + ¯ a + ¯ a − ¯ a .
2 2
¯
A simple example of a symmetric dyadic is the unit dyadic I defined by
ˆ ˆ
ˆ ˆ
ˆ ˆ
¯ I = i 1 i 1 + i 2 i 2 + i 3 i 3 .
This quantity often arises in the manipulation of dyadic equations, and satisfies
¯
¯
A · I = I · A = A
¯
for any vector A. In matrix form I is the identity matrix:
 
100
¯
[I] =  010  .
001
The components of a dyadic may be complex. We say that ¯ a is hermitian if
∗
B · ¯ a = ¯ a · B (A.72)
T
∗
holds for any B. This requires that ¯ a = ¯ a . Taking the transpose we can write
∗ T
¯ a = (¯ a ) = ¯ a †
where “†” stands for the conjugate-transpose operation. We say that ¯ a is anti-hermitian
if
B · ¯ a =−¯ a · B (A.73)
∗
T
∗
for arbitrary B. In this case ¯ a =−¯ a . Any complex dyadic can be decomposed into
hermitian and anti-hermitian parts:
1 H A
¯ a = ¯ a + ¯ a (A.74)
2
where
H
A
†
†
¯ a = ¯ a + ¯ a , ¯ a = ¯ a − ¯ a . (A.75)
A dyadic identity important in the study of material parameters is
∗
∗
∗
†
B · ¯ a · B = B · ¯ a · B. (A.76)
We show this by decomposing ¯ a according to (A.74), giving

1 H ∗ A ∗

∗ ∗ ∗ ∗ ∗
B · ¯ a · B = B · ¯ a + B · ¯ a · B
2
∗
∗
where we have used (B · ¯ a) = (B · ¯ a ). Applying (A.72) and (A.73) we obtain
∗

1 ∗ ∗ ∗ ∗

∗ ∗ H∗ A∗ ∗
B · ¯ a · B = ¯ a · B − ¯ a · B · B
2
∗ 1 H A
= B · ¯ a · B − ¯ a · B
2

1 H A
∗
= B · ¯ a − ¯ a · B .
2
A
H
Since the term in brackets is ¯ a − ¯ a = 2¯ a by (A.75), the identity is proved.
†
© 2001 by CRC Press LLC