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11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 211
vector form:
⎛
⎞
M 0 (ρ 2 ) 0
M 1 (ρ 2 ) M 0 (ρ 2 )
. M 2 (ρ 2 ) M 1 (ρ 2 ) ..
⎛
. . M 2 (ρ 2 ) .. M0 (ρ 2 )
⎜
MÑ(ρ 2 )
⎝
. M 1 (ρ 2 )
MÑ (ρ 2 ) M 2 (ρ 2 )
⎜
.
⎝
..
. ..
⎟
⎠
0 MÑ(ρ 2 )
⎛
M η1,η 2
(ρ 2 ) ⎜
⎝
z 0 (ρ 2 )
−z 1 (ρ 2 )
.
(−1) η1 z η1 (ρ 2 )
z 0 (ρ 2 )
−z 1 (ρ 2 )
.
(−1) η1 z η1 (ρ 2 )
⎞
⎟
⎠ =
(11.44)
⎞
⎟
⎠ =0
where the block matrix M η1,η 2
(ρ 2 ) has dimension (η 1 + Ñ +1) m × (η 1 +1) n.
Furthermore it holds that the rank r η1,η 2
< (η 1 +1)n.
Because η 2 denotes the degree of z(ρ 1 ,ρ 2 ) with respect to ρ 2 , we can write:
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
(11.45)
for i = 0,...,η 1 . When (11.45) is substituted into the equations of (11.44) and
the matrices M i,j in Equation (11.41) are used, the following system of equations in
matrix-vector form is constructed:
⎛ ⎞
z 0,0
.
z 0,η2
M η1,η 2 .
=0, (11.46)
z η1,0
⎜ . ⎟
⎝ . ⎠
z η1,η 2
where the matrix M η1,η 2
is a matrix of (η 2 + Ñ + 1)(η 1 + Ñ +1)m rows and (η 1 +
1)(η 2 +1) n columns containing the m × n coefficient matrices M i,j from Equation
(11.41).
Example 11.1 (continued). To illustrate the construction of the matrix M η1,η 2
, using
this approach, the example of the previous subsection is continued.
Suppose that the two-parameter matrix pencil A(ρ 1 ,ρ 2 ) T of dimension m × n is
given where the total degree of the terms is Ñ = 3 and one wants to check whether