For completeness, we consider a pencil with all different types
of structure blocks in its KCF:
Since these forms are computed using the staircase reduction, the block indices and start to count from the southeast corner. Now, superdiagonal blocks of and have full row rank and diagonal blocks of and have full column rank. In the following table, the structure indices for the , , and staircase forms are summarized.



So far the description for computing the GUPTRI form has relied on infinite precision arithmetic. In the presence of roundoff the problem is regularized by allowing a deflation criterion for range/null space separations and thereby makes it possible to compute the GUPTRI form of a nearby matrix pencil.
This GUPTRI form is computed by a sequence of unitary equivalence transformations. The equivalence transformations are built from rankrevealing factorizations used to find orthonormal bases for different null spaces associated with the matrix pair.