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
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:
iAAAAAAA has all right singular structure (the right minimal indices).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 .
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).
in GUPTRI form we also know different pairs of
reducing subspaces [451,121].
Suppose the eigenvalues on the diagonal of
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