link to my thesis

11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 199
where the matrices B, C, and D have at most ((N − 1) 2 + 1)2 N unique rows and
exactly (N −1) 2 2 N columns. Let us denote the dimensions of these matrices by m×n.
The equivalence of (11.7) and (11.3) implies that (11.7) has a finite number of
solutions too.
Let us now consider the rectangular eigenvalue problem (11.7) as a one-parameter
eigenvalue problem in ρ 1 with matrices containing entries from the field C(ρ 2 ). Then
(11.7) can be rewritten as the equivalent homogeneous system of equations:
(
P (ρ2 )+ρ 1 Q(ρ 2 ) ) w = 0 (11.8)
where P (ρ 2 )=B + ρ 2 D and Q(ρ 2 )=C, both of dimensions m × n.
Because the one-parameter pencil P + ρ 1 Q in problem (11.8) is rectangular it
is singular by definition. We now could proceed as in Section 10.3: by computing
the Kronecker Canonical form of this one-parameter matrix pencil and by splitting
off its singular parts, one can try to determine the regular part of the pencil. Then
one expects to compute solutions ρ 1 and ρ 2 of the original problem (11.3) from this
regular part.
The next subsections describe the transformation into the Kronecker canonical
form, the structure of the involved transformed matrices and three methods to determine
these transformation matrices to split off singular parts using exact arithmetic.
We use here in fact a generalization of the techniques of the Kronecker canonical form
computations for singular one-parameter pencils in C presented in Section 10.3 of the
previous chapter.
11.3.1 Transformation into the Kronecker canonical form
When varying the value of ρ 1 in Equation (11.8) it may happen that solutions exist
for all values of ρ 1 , but other solutions may exist only for special values of ρ 1 . To
emphasize the dependence of the solution set on the value of ρ 1 , we shall write w(ρ 1 ).
The matrix pencil P + ρ 1 Q can be brought into Kronecker canonical form. This
involves a transformation into the form:
W −1 (P + ρ 1 Q)V = D + ρ 1 E (11.9)
where V (of size n × n) and W (of size m × m) are invertible transformation matrices
with entries in C(ρ 2 ). The solutions ˜w(ρ 1 ) to the set of equations:
(D + ρ 1 E)˜w = 0 (11.10)
are related to those of the original problem (P + ρ 1 Q)w = 0 according to:
w(ρ 1 )=V ˜w(ρ 1 ). (11.11)
Note that the systems of equations are equivalent when working over C(ρ 2 ), i.e.,
every solution ˜w(ρ 1 ) corresponds to a unique solution w(ρ 1 ) and vice versa, since V
is invertible.