Generic and Nongeneric Kronecker Structures

Although the KCF looks quite complicated in the general case, most matrix pencils have a more simple Kronecker structure. If $A - \lambda B$ is $m \times n$, where $m \neq n$, then for almost all $A$ and $B$ it will have the same KCF, depending only on $m$ and $n$. This corresponds to the generic case when $A - \lambda B$ has full rank for any complex (or real) value of $\lambda$. Accordingly, generic rectangular pencils have no regular part. The generic Kronecker structure for $A - \lambda B$ with $d = n - m > 0$ is diag(L_, ..., L_, L_+ 1, ..., L_+ 1), where $\ell = \lfloor m/d \rfloor$, the total number of blocks is $d$, and the number of $L_{\ell + 1}$ blocks is $m~{\rm mod}~d$ (which is 0 when $d$ divides $m$) [446,116]. The same statement holds for $d = m - n > 0$ if we replace $L_{\ell}, L_{\ell + 1}$ in (8.31) by $L_{\ell}^T, L_{\ell + 1}^T$. For example, a generic pencil of size $2 \times 3$ has an $L_2$ block as its KCF.

Square pencils are generically regular; i.e., ${\rm det}(A - \lambda B) = 0$ if and only if $\lambda$ is an eigenvalue. Moreover, the most generic regular pencil is diagonalizable with distinct finite eigenvalues. The generic singular pencils of size $n \times n$ have the Kronecker structures [456]: diag(L_j, L_n-j-1^T),     j = 0, ..., n-1. Only if a singular $A - \lambda B$ is rank deficient (for some $\lambda$), 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].