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178 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
Formally, when the regular pencil D r + ρ 1 E r in (10.17) attains the canonical form
(10.20), then the Kronecker canonical form of the pencil B + ρ 1 C is obtained.
The Jordan blocks N j in (10.20) are square blocks of possibly different dimensions
k j × k j with a structure as in (10.21). The equations resulting from N j w 1 = 0, where
w 1 is the corresponding part of the eigenvector w, are:
⎧
w 1,1 + ρ 1 w 1,2 =0
⎪⎨
w 1,2 + ρ 1 w 1,3 =0
.
w 1,kj−1 + ρ 1 w 1,kj =0
⎪⎩
w 1,kj =0
These equations only admit the trivial solution w 1 = 0. Therefore, ρ 1 = ∞ does
not play a role in our problem. The blocks N j are said to generate infinite eigenvalues.
The number of infinite eigenvalues amounts to the sum of the associated dimensions
k j for these blocks.
The Jordan blocks J j in (10.20) are square blocks of possibly different dimensions
l j × l j with a structure as in (10.22). The corresponding equations following from
J j w 1 = 0, are: ⎧⎪ ⎨
⎪ ⎩
ρ 1 w 1,1 + w 1,2 =0
ρ 1 w 1,2 + w 1,3 =0
.
ρ 1 w 1,lj−1 + w 1,lj =0
ρ 1 w 1,lj =0
In this case there are two possible situations: if ρ 1 0, then J j w 1 only admits the
trivial solution. If ρ 1 = 0 it admits the non-trivial solution w 1 =(1, 0,...,0) T ,orany
vector proportional to it. This block corresponds to a zero eigenvalue with algebraic
multiplicity l j .
The Jordan blocks corresponding to non-zero finite eigenvalues are denoted by
the R j blocks of dimension m j × m j in Equation (10.20). They exhibit the structure
as in (10.23). The finite eigenvalues have the values ρ 1 = α j with multiplicity equal
to the sum of the associated dimensions m j , for each value of α j . The corresponding
equations resulting from R j w 1 = 0 are:
⎧
(ρ 1 − α j )w 1,1 + w 1,2 =0
⎪⎨
(ρ 1 − α j )w 1,2 + w 1,3 =0
.
(ρ 1 − α j )w 1,mj−1 + w 1,mj =0
⎪⎩
(ρ 1 − α j )w 1,mj =0