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 : 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].