Resonances in Repetitive Systems
277
FIGURE 11.1: Distinct eigenvalues around the unit circle. Small circles
show the neighborhood of each eigenvalue.
Let us focus on coupling between horizontal and vertical planes. The matrix
ˆ
M is a 4 × 4 symplectic matrix, which we describe as
ˆ
M =
ˆ
A ˆ
B
ˆ
C ˆ
D
,
where ˆ
A, ˆ
B, ˆ
C and ˆ
D are 2 × 2 matrices. From eq. (5.11), we have
ˆ
M
−1 =
¯
A ¯
C
¯
B ¯
D
,
where ¯
A is defined as ¯
A = − ˆ
J ˆ
A
T ˆ
J, as well as ¯
B, ¯
C and ¯
D. Let us solve for
the eigenvalue Λ of ˆ
M + ˆ
M
−1 . Obviously Λ = λ + 1/λ, because
ˆ
MM v = λλ v =⇒ v = λ ˆ
M
−1 v =⇒
1
λ
v = ˆ
M
−1
v
=⇒
ˆ
M + ˆ
M
−1
v =
λ +
1
λ
v.
ˆ
M + ˆ
M
−1 =
ˆ
A ˆ
B
ˆ
C ˆ
D
+
¯
A ¯
C
¯
B ¯
D
=
ˆ
A + ¯
A ˆ
B + ¯
C
ˆ
C + ¯
B ˆ
D + ¯
D
.
Note that
ˆ
A + ¯
A =
a 11 a 12
a 21 a 22
+
a 22 −a 12
−a 21 a 11
= tr ˆ
A ·
1 0
0 1
= tr ˆ
A · ˆ
I.
Using the relation
ˆ
M + ˆ
M
−1
=
tr ˆ
A · ˆ
I ˆ
B + ¯
C
ˆ
C + ¯
B tr ˆ
D · ˆ
I
,