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11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 207
resulting in a smaller sized pencil ˜P (ρ2 )+ρ 1 ˜Q(ρ2 ), still containing all the relevant
(meaningful) solutions. If there are more than one solution z(ρ 1 ,ρ 2 ) of degrees η 1
and η 2 , let us say g solutions, then g singular blocks of dimensions (η 1 +1)× η 1 can
be split off at once, provided that independence for the associated basis holds.
The transformation of P (ρ 2 )+ρ 1 Q(ρ 2 ) into a quasi diagonal pencil (11.31) can
be done using a generalization of the transformation technique mentioned in Section
10.3.1 which is based on choosing appropriate transformation matrices V and W .
Consider the pencil P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T of dimension m × n as a family of operators
mapping C n (ρ 2 )toC m (ρ 2 ). With a suitable choice of bases in the spaces the matrix
corresponding to the operator P (ρ 2 )+ρ 1 Q(ρ 2 ) takes the form (11.31).
Such a suitable choice corresponds to taking the vectors z 0 (ρ 2 ), ..., z η1 (ρ 2 )ofa
solution z(ρ 1 ,ρ 2 ) of (11.22) as a part of the basis for C n (ρ 2 ) and P (ρ 2 ) T z 1 (ρ 2 ), ...,
P (ρ 2 ) T z η1 (ρ 2 ) as a part of the basis for C m (ρ 2 ). This is allowed since {z 0 (ρ 2 ), ...,
z η1 (ρ 2 )} and {P (ρ 2 ) T z 1 (ρ 2 ), ..., P(ρ 2 ) T z η1 (ρ 2 )} are linearly independent sets. If
the bases for C m (ρ 2 ) and C n (ρ 2 ) are completed with standard basis column vectors
w η1+1,...,w m and v η1+2,...,v n to make W and V square and invertible, then the
transformation:
W −1 (ρ 2 )
(
P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T ) V (ρ 2 ) (11.32)
yields the equivalent pencil (11.31), up to the upper right zero block. To make the
upper right block also zero, the columns w η1+1,...,w m and v η1+2, ..., v n should be
chosen in a special way.
In order to get rid of all the singular parts L T η i
of the matrix pencil, one has to
search for solutions z(ρ 1 ,ρ 2 ) for different values of η 1 and η 2 . We have previously
required η 1 to be minimal first, and (given η 1 ) η 2 to be minimal next. Once one
or more such solutions z(ρ 1 ,ρ 2 ) are found, one has to construct the transformation
matrices W (ρ 2 ) and V (ρ 2 ) and to apply these as in (11.32). This can be repeated
iteratively until no more singular parts exist. Note that the parts that are split
off increase in size in every step which makes it possible to put limitations on the
dimensions of the next singular blocks that can be found. E.g., if m × n is currently
reduced to 14 × 12 and η 1 = 4 is exhausted, we know that (i) 14 − 12 = 2 more
blocks will be found and (ii) these are of dimension 6 × 5 and 8 × 7 or two blocks of
dimension 7 × 6, since all other options are no longer available.
Example 11.1 (continued). In Example 11.1 the value of η 1 is chosen as 1 and an
approach is given to compute the vectors z 0 (ρ 2 ) and z 1 (ρ 2 ) which satisfy the equations
in (11.28). To transform the pencil P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T into the equivalent form