link to my thesis

11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 215
Example 11.1 (continued). The transformation matrices for the running example
using this approach are constructed as follows. The matrix V is structured as:
⎛
⎞
V (ρ 2 )=
z 0 (ρ 2 ) z 1 (ρ 2 ) v 3 ... v n . (11.54)
⎜
⎟
⎝
⎠
The transformation matrix W contains only one prescribed column. Equation (11.52)
provides for c = 0 an expression for the first column of this matrix:
M 0 (ρ 2 )z 1 (ρ 2 ) − ρ 1 M 2 (ρ 2 )z 0 (ρ 2 ) − ρ 2 1M 3 (ρ 2 )z 0 (ρ 2 ). (11.55)
By following the Equations in (11.49), it turns out that this is equal to:
(
)
M 0 (ρ 2 ) − ρ 1 M 1 (ρ 2 ) − ρ 2 1M 2 (ρ 2 ) z 1 (ρ 2 ), (11.56)
which results in the following matrix W (ρ 2 ):
⎛
⎞
(
)
W (ρ 2 )=
M 0 (ρ 2 ) − ρ 1 M 1 (ρ 2 ) − ρ 2 1M 2 (ρ 2 ) z 1 (ρ 2 ) w 2 ... w m . (11.57)
⎜
⎟
⎝
⎠
The columns v 3 ,...,v n ,andw 2 ,...,w m denote standard basis vectors to make the
matrices V and W square and invertible.
When the transformation:
W (ρ 1 ,ρ 2 ) −1( )
M 0 (ρ 2 )+ρ 1 M 1 (ρ 2 )+ρ 2 1M 2 (ρ 2 )+ρ 3 1M 3 (ρ 2 ) V (ρ 2 ) (11.58)
is applied, an equivalent pencil emerges and a similar transposed version of the singular
part L T η 1
of dimension 1 × 2 as in (11.37) can be split off from the pencil.
11.3.4 A second approach to compute the transformation matrices for a
non-linear two-parameter matrix pencil
This section presents a second approach to split off singular parts of a polynomial/
non-linear matrix pencil. This approach differs from the approach described in the
previous section in the way the expansion of the involved polynomial matrices is performed.
Although this approach will not work in general, it exhibits some useful