Kronecker Canonical Form

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 [187].
Suppose
.
Then there exist nonsingular
and
such that
P^-1 (A - B)Q = Ã - B
diag( A_1 - B_1 , ..., A_b -
B_b ) ,
where
is
.
We can partition the columns of and into
blocks corresponding to the diagonal
blocks of
:

where is , and is , such that

Each block must be of one of the following forms:

First we consider J_j ( ) cccc - & 1 & &

& & &

& & & 1

& & & - and N_j cccc 1 & -& &

& & &

& & & -

& & & 1 . is simply a Jordan block, and is called a

span the range. From the cyclicity of the transformation we get the vector chain relations

The vector is said to be a

is a block
corresponding to an *infinite eigenvalue* of multiplicity :

By exchanging the -part and -part of we get a block. So if the KCF of has an block in its KCF, then has a block in its KCF and vice versa. This fact is utilized in algorithms for computing canonical structure of matrix pencils.

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
L_j cccc
- & 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 .