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11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 213
1) n =30m × 6n:
⎛
M 1,2 =
⎜
⎝
M 0,0 0 0 0 0 0
M 0,1 M 0,0 0 0 0 0
M 0,2 M 0,1 M 0,0 0 0 0
M 0,3 M 0,2 M 0,1 0 0 0
0 M 0,3 M 0,2 0 0 0
0 0 M 0,3 0 0 0
M 1,0 0 0 M 0,0 0 0
M 1,1 M 1,0 0 M 0,1 M 0,0 0
M 1,2 M 1,1 M 1,0 M 0,2 M 0,1 M 0,0
0 M 1,2 M 1,1 M 0,3 M 0,2 M 0,1
0 0 M 1,2 0 M 0,3 M 0,2
0 0 0 0 0 M 0,3
M 2,0 0 0 M 1,0 0 0
M 2,1 M 2,0 0 M 1,1 M 1,0 0
0 M 2,1 M 2,0 M 1,2 M 1,1 M 1,0
0 0 M 2,1 0 M 1,2 M 1,1
0 0 0 0 0 M 1,2
0 0 0 0 0 0
M 3,0 0 0 M 2,0 0 0
0 M 3,0 0 M 2,1 M 2,0 0
0 0 M 3,0 0 M 2,1 M 2,0
0 0 0 0 0 M 2,1
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 M 3,0 0 0
0 0 0 0 M 3,0 0
0 0 0 0 0 M 3,0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
⎞
. (11.51)
⎟
⎠
Note that the zero rows are redundant here and may be removed from the matrix
M 1,2 .
If the rank of this matrix M 1,2 is smaller than 6n, one can compute all solutions
z(ρ 1 ,ρ 2 ) of A(ρ 1 ,ρ 2 ) T z = 0: every non-zero vector in the kernel of this
matrix yields a solution z(ρ 1 ,ρ 2 ). Every vector in the kernel has the structure
[z0,0,z T 0,1,z T 0,2,z T 1,0,z T 1,1,z T 1,2] T T and one can construct the solution z(ρ 1 ,ρ 2 ) from it
as: z(ρ 1 ,ρ 2 )=(z 0,0 + ρ 2 z 0,1 + ρ 2 2 z 0,2 ) − ρ 1 (z 1,0 + ρ 2 z 1,1 + ρ 2 2 z 1,2 ).
In this approach, the structure of the matrix V (ρ 2 ) is unchanged: the first columns
of V (ρ 2 ) are chosen (as before) as the vectors z 0 (ρ 2 ),...,z η1 (ρ 2 ). However, the first
columns of the transformation matrix W are chosen in a different way. The η 1 first
columns of W (ρ 1 ,ρ 2 ) are denoted by w 0 ,...,w η1−1 and are given by the following