Generalized Schur-Staircase Form

In general, we cannot guarantee stable computation of the KCF of a pencil, since
the transformation matrices that reduce to KCF can be
arbitrarily ill-conditioned.
However, it is possible to compute the Kronecker structure (or parts of it)
using only unitary transformations.
The price we have to pay is a denser canonical form, called a
*generalized Schur-staircase form*.
This form is block upper triangular with diagonal blocks
in staircase form (also block upper triangular) that reveal the
fine structure elements of the KCF.

In most applications it is enough to reduce to a
generalized Schur-staircase form, e.g., to GUPTRI form
[121,122]:
P^ (A- B) Q =
ccc A_r - B_r & * & *

0 & A_reg - B_reg & *

0 & 0 & A_l - B_l ,
where () and () are unitary.
Here the square upper triangular block
is regular and has the same regular structure as
(i.e., contains all finite and infinite eigenvalues
of ).
The rectangular block
has only right minimal indices in its
KCF--indeed the same blocks as .
Similarly,
has only left minimal indices in its KCF,
the same blocks as .
If is
singular, at least one of
and
will be present in (8.34).
If is regular,
and
are not present in (8.34) and the GUPTRI form reduces to
.
Staircase forms that reveal the Jordan structure of the
zero and infinite eigenvalues are contained in
:
A_reg = ccc
A_z & * & *

0 & A_f & *

0 & 0 & A_i ,
B_reg = ccc
B_z & * & *

0 & B_f & *

0 & 0 & B_i .

In summary, the diagonal blocks of the GUPTRI form of describe the Kronecker structure as follows:

The explicit structure of the diagonal blocks in staircase form is presented in the next section. The nonzero, finite eigenvalues of (if any) are in the block but their multiplicities or Jordan structures are not computed explicitly. However, it is possible to extract the Jordan structure of a finite, nonzero eigenvalue of by computing the (right-zero)-staircase form (see §8.7.6) of the shifted pencil , which has zero as an eigenvalue of multiplicity .iAAAAAAA has all right singular structure (the right minimal indices).

has all Jordan structure for the zero eigenvalue.

has all finite, nonzero eigenvalues.

has all Jordan structure for the infinite eigenvalue.

has all left singular structure (the left minimal indices).

Given
in GUPTRI form we also know different pairs of
*reducing subspaces* [451,121].
Suppose the eigenvalues on the diagonal of
are
ordered so that the first , say, are in
(a subset of the spectrum) and the remainder are outside .
Then the GUPTRI form can also be expressed as
P^ (A - B) Q =
cc A_11 - B_11 & A_12 - B_12

0 & A_22 - B_22,
where
contains
and the regular part corresponding to
, and
contains
the remaining regular part and
.
If
is
,
then the left and right reducing subspaces corresponding to are
spanned by the leading columns of (denoted ) and
the leading columns of (denoted ), respectively,
such that

When is empty, the corresponding reducing subspaces are called