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11.5. EXAMPLE 225
of both matrices simultaneously. The other intersections of both curves correspond
to the trivial solutions x 1 =0,..., x N = 0. Figure 11.3 shows a zoomed version of
Figure 11.2 near these 5 real points which are marked with green dots.
Figure 11.3: Five important real eigenvalue pairs (ρ 1 ,ρ 2 ) of the matrices
ÃãN−1 (ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T
Splitting off singular blocks from the matrix A 1 (ρ 1 ,ρ 2 ) T is done by performing transformations
W −1 A 1 (ρ 1 ,ρ 2 ) T V with suitable square and invertible matrices V and
W . When performing these transformations iteratively to the matrix A 1 (ρ 1 ,ρ 2 ) T ,
one can split off one or more singular parts in every iteration.
Finally, one has a set of transformation matrices at hand. All the values ρ 2 (and
possibly ρ 1 ) which make the matrices V and W singular are the (partial) solutions
of the system of equations (11.78). The results of these reduction processes are
given in the Tables 11.1 and 11.2 for the approaches of Subsections 11.3.3 and 11.3.4,
respectively. The tables show the dimensions of the matrices, the degrees η 1 and η 2 of
the solution z(ρ 1 ,ρ 2 ), the dimensions of the transposed version of the singular parts
L T η 1
which are split off and the variables involved in the matrices.
Table 11.1: Results of reduction process of two-parameter pencil A 1 (ρ 1 ,ρ 2 ) T using
the approach of Subsection 11.3.3.
Matrix Dimensions η 1 η 2 Dimensions of L η1 Variables
A 1 (ρ 1 ,ρ 2 ) T 15 × 30 0 0 4(0× 1) ρ 1 ,ρ 2
A 2 (ρ 1 ,ρ 2 ) T 15 × 26 1 1 6(1× 2) ρ 1 ,ρ 2
A 3 (ρ 1 ,ρ 2 ) T 9 × 14 5 4 5 × 6 ρ 1 ,ρ 2
A 4 (ρ 1 ,ρ 2 ) T 4 × 8 4 7 4 × 5 ρ 1 ,ρ 2
A 5 (ρ 1 ,ρ 2 ) T 0 × 3 − − − −