Although the KCF looks quite complicated in the general case, most
matrix pencils have a more simple Kronecker structure.
If is , where , then for almost all
and it will have the same KCF, depending only on and .
This corresponds to the *generic case* when has full rank
for any complex (or real) value of .
Accordingly, generic rectangular pencils have no regular part.
The generic Kronecker structure for with
is
diag(L_, ..., L_, L_+ 1,
..., L_+ 1),
where
, the total number of blocks is , and
the number of blocks is (which is 0 when
divides ) [446,116].
The same statement holds for
if we replace
in (8.31) by
.
For example, a generic pencil of size has
an block as its KCF.

Square pencils are generically regular; i.e.,
if and only if is an eigenvalue.
Moreover, the most generic regular pencil is diagonalizable
with distinct finite eigenvalues.
The generic singular pencils of size have the
Kronecker structures [456]:
diag(L_j, L_n-j-1^T), j = 0, ..., n-1.
Only if a singular is rank deficient (for some ),
the associated KCF may be more complicated and possibly include a regular part,
as well as right and left singular blocks.
This situation corresponds to the *nongeneric* (or *degenerate*)
*case*, which is the real challenge from a computational point of view.
Degenerate rectangular pencils have several applications in
control theory, for example, to compute the
controllable subspace and uncontrollable modes
of a linear descriptor system [447,120].