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
:
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
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
.