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11.3. KRONECKER CANONICAL FORM OF A TWO-PARAMETER PENCIL 203
Now problem (11.8) is reshaped into the form of two one-parameter eigenvalue
problems, which can be solved by using the techniques presented in Chapter 10. The
main goal now is to determine the transformation matrices V (ρ 2 ) and W (ρ 2 ) which
bring the singular pencil P +ρ 1 Q into the Kronecker canonical form. This is the main
subject of the next three subsections where three methods are described to determine
these matrices V (ρ 2 ) and W (ρ 2 ).
Remark 11.1. The only singular blocks that may be split off from the singular pencil
P +ρ 1 Q, are the blocks L T η of dimension (η +1)×η, see Proposition 11.1. Such blocks
can be obtained by looking at transformations:
W (ρ 2 ) −1 (P T + ρ 1 Q T )V (ρ 2 ), (11.21)
which means that we continue to work with the transposed versions of the matrices
P and Q. A consequence of this is that we find as singular blocks the transposes of
the blocks L T η which then are of dimensions η × (η +1).
11.3.2 Computing the transformation matrices for a linear two-parameter
matrix pencil
We consider the transposed version of the pencil in Equation (11.8). The dimensions
of the matrices P (ρ 2 ) T and Q(ρ 2 ) T are m×n. Because the rectangular one-parameter
pencil P (ρ 2 ) T + ρ 1 Q(ρ 2 ) T is singular, the rank r will be r