link to my thesis

11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 201
and where J is the Jordan form of a κ × κ square matrix pencil which is regular and
only has finite eigenvalues.
As seen before in the previous chapter, every matrix pencil P + ρ 1 Q has a
unique Kronecker canonical form, provided that the dimensions ε 1 ≤ ε 2 ≤··· ≤ε s ,
η 1 ≤ η 2 ≤ ··· ≤η t , υ 1 ≤ υ 2 ≤ ··· ≤υ u and the Jordan blocks J are ordered in some
prespecified way.
A solution to the problem (D + ρ 1 E)˜w = 0 involves a partitioned eigenvector ˜w
structured as:
⎛
⎞
˜w 0
˜w 1
. .
˜w s
˜w s+1
˜w =
.
. (11.16)
˜w s+t
˜w s+t+1
.
⎜
⎟
⎝ ˜w s+t+u ⎠
˜w s+t+u+1
Each of the parts in ˜w may yield a solution, to be combined with solutions to other
parts. Each part will always admit the trivial solution. Furthermore, it holds that:
• ˜w 0 is always free to choose, irrespective of ρ 1 .
• ˜w i , i =1,...,s needs to satisfy L εi ˜w i = 0. Hence,
˜w i = α i
⎛
⎜
⎝
1
−ρ 1
ρ 2 1
.
.
(−ρ 1 ) εi
for arbitrary α i ∈ C(ρ 2 ), and ρ 1 is free to choose.
⎞
, (11.17)
⎟
⎠
• ˜w s+i , i =1,...,t, needs to satisfy L T η i ˜w s+i = 0, which only admits the trivial
solution ˜w s+i =0.
• ˜w s+t+i , i =1,...,u, needs to satisfy N υi ˜w s+t+i = 0, which only admits the
trivial solution ˜w s+t+i =0.
• ˜w s+t+u+1 needs to satisfy J ˜w s+t+u+1 = 0. For given ρ 2 , only a finite number
of values for ρ 1 will do, i.e., the eigenvalues which are finite.