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11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 217
z(ρ 1 ,ρ 2 ) into (11.62) where the matrix A(ρ 1 ,ρ 2 ) T is expanded as in (11.59) leads to:
(
B(ρ 2 )+ρ 1 C(ρ 1 ,ρ 2 )) (z0
(ρ 2 ) − ρ 1 z 1 (ρ 2 )+ρ 2 1z 2 (ρ 2 )+...+(−1) η1 ρ η1
1 z η 1
(ρ 2 ) ) =
(
1 ·
(
ρ 1 ·
(
ρ 2 1 ·
.
(
ρ η1
1 ·
(
ρ η1+1
1 ·
+B(ρ 2 ) z 0 (ρ 2 )
−B(ρ 2 ) z 1 (ρ 2 ) +C(ρ 1 ,ρ 2 ) z 0 (ρ 2 )
+B(ρ 2 ) z 2 (ρ 2 ) −C(ρ 1 ,ρ 2 ) z 1 (ρ 2 )
.
.
(−1) η1 B(ρ 2 ) z η1 (ρ 2 ) (−1) η1−1 C(ρ 1 ,ρ 2 ) z η1−1(ρ 2 )
(−1) η1 C(ρ 1 ,ρ 2 ) z η1 (ρ 2 )
)
)
)
)
)
+
+
+
.
+
=0.
(11.64)
One can now work out the relationships for the coefficients of the powers of ρ 1
and write all the resulting expressions in matrix-vector form, as follows:
⎛
⎜
⎝
B(ρ 2 ) 0 ... ... 0
C(ρ 1 ,ρ 2 ) B(ρ 2 )
.
0 C(ρ 1 ,ρ 2 )
. .. . ..
0 0
. .. . .. . ..
.
.
. .. B(ρ2 )
0 0 ... ... C(ρ 1 ,ρ 2 )
⎞
⎛
⎜
⎝
⎟
⎠
z 0
−z 1
z 2
.
(−1) η1 z η1
⎞
=
⎟
⎠
(11.65)
⎛
M η1,η 2
(ρ 1 ,ρ 2 )
⎜
⎝
z 0
−z 1
z 2
.
⎞
=0.
⎟
⎠
(−1) η1 z η1
Here the block matrix M η1,η 2
(ρ 1 ,ρ 2 ) has dimension (η 1 +2) m × (η 1 +1) n. The rank
of the matrix M η1,η 2
(ρ 1 ,ρ 2 )isr η1,η 2
< (η 1 +1)n.
Because η 2 is the degree of ρ 2 in z(ρ 1 ,ρ 2 ), we can write for z i (ρ 2 ):
z i (ρ 2 )=z i,0 + ρ 2 z i,1 + ρ 2 2 z i,2 + ...+ ρ η2
2 z i,η 2
, (11.66)
for i =0,...,η 1 .
When (11.66) is substituted into (11.65), the matrices B(ρ 2 ) and C(ρ 1 ,ρ 2 ) are
written as in (11.60) and (11.61) and the relationships for the coefficients of the