Just as the Jordan canonical form (JCF) describes the invariant subspaces and eigenvalues of a square matrix in full detail, there is a Kronecker canonical form (KCF) which describes the generalized eigenvalues and generalized eigenspaces of a pencil in full detail. In addition to Jordan blocks for finite and infinite eigenvalues, the Kronecker form contains singular blocks corresponding to minimal indices of a singular pencil (see below).
The KCF of exhibits the fine structure elements,
including elementary divisors (Jordan blocks) and minimal indices
(singular blocks), and is defined as follows .
Then there exist nonsingular
P^-1 (A - B)Q = Ã - B
diag( A_1 - B_1 , ..., A_b -
B_b ) ,
We can partition the columns of and into
blocks corresponding to the diagonal
is a block
corresponding to an infinite eigenvalue of multiplicity :
The and blocks together constitute the regular structure of the pencil. All the are regular blocks if and only if is a regular pencil. denotes the eigenvalues of the regular part of (counted with multiplicities) and is called the spectrum of .
The other two types of diagonal blocks are
- & 1 & &
& & &
& & - & 1 and L_j^T ccc -& &
1 & &
& & -
& & 1 . The block is called a singular block of right (or column) minimal index . It has a one-dimensional right null space, , for any , i.e.,
Similarly, the block is a singular block of left (or row) minimal index and has a one-dimensional left null space for any . The left and right singular blocks together constitute the singular structure of the pencil and appear in the KCF if and only if the pencil is singular. The regular and singular structures define the Kronecker structure of a singular pencil.
We end this introductory description by briefly pointing to the relationship between structure information of the KCF and the GUPTRI form (8.28). The block contains all information about the right singular blocks and contains all information about the left singular blocks. The regular part corresponds to .