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202 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
We can conclude from this that non-trivial solutions only exist for a finite set of
values for ρ 1 if and only if g = 0 and s = 0 in the Kronecker canonical form (11.12)
of the singular matrix pencil P + ρ 1 Q in (11.9) (thus no blocks of the form L ε ).
The dimension of the regular part involving the block J is κ × κ. This means that
there are κ equations in κ − 1 unknowns in the (normalized) eigenvector ˜w s+t+u+1
plus the two unknowns ρ 1 and ρ 2 in the eigenvalue problem J(ρ 1 ,ρ 2 )˜w s+t+u+1 .This
means that we have created an underdetermined system of equations. Such a system
allows an infinite number of additional solutions. This is in contradiction of what we
know when we apply the Stetter-Möller matrix method to the problem at hand: the
commuting operators Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T will only allow for a finite
number of solutions. Therefore one may conclude that generically there will exist no
regular part in this pencil, thus κ =0.
Therefore, a finite number of solutions for ρ 1 and ρ 2 may only exist, if and only
if, g =0,s = 0, and κ = 0 and we conclude the following:
Proposition 11.1. The pencil P + ρ 1 Q with P, Q ∈ C(ρ 2 ) m×n has a finite set of
pairs (ρ 1 ,ρ 2 ) at which the two-parameter eigenvalue problem (P + ρ 1 Q)w =0admits
a non-trivial solution if and only if its Kronecker canonical form over C(ρ 2 ) only
contains singular blocks:
⎛
⎞
ρ 1 0 ... 0
. 1 ρ ..
1 L T η =
. 0 1 .. 0
(11.18)
⎜
⎝
.
. .. ⎟
ρ1 ⎠
0 ... 0 1
(possibly of size 1 × 0) which correspond to indeterminate eigenvalues and blocks of
the form:
⎛
⎞
1 ρ 1 0
. .. . .. N υ = ⎜
⎟
(11.19)
⎝
ρ 1
⎠
0 1
which correspond to infinite eigenvalues.
As a consequence, the generic solution set for D + ρ 1 E, when considered over the
field C(ρ 2 ), is just the trivial set ˜w(ρ 1 ) = 0. Hence, we have the following result:
Proposition 11.2. The finite solutions for (ρ 1 ,ρ 2 ) correspond to values for ρ 2 which
make the transformation matrices V and W singular (as numerical values). Hence,
we arrive at the one-parameter problem to investigate the existence of non-trivial
solutions to: ⎧
⎨ V (ρ 2 )z v =0 of size m × m
(11.20)
⎩
W (ρ 2 )z w =0 of size n × n.