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10.3. THE KRONECKER CANONICAL FORM OF A MATRIX PENCIL 171
Example 10.1. As an illustrative example, the Kronecker canonical form of a pencil
B + ρ 1 C with h =2,g =1,ε 1 =2,ε 2 =3,η 1 = 1 and η 2 = 2 is shown below:
⎛
⎞
0
0
ρ 1 1 0
0 ρ 1 1
ρ 1 1 0 0
0 ρ 1 1 0
0 0 ρ 1 1
. (10.24)
ρ 1
1
ρ 1 0
1 ρ 1
⎜
⎟
⎝
0 1
⎠
D r + ρ 1 E r
The empty blocks correspond to zero blocks of appropriate dimensions. The pencil
D r + ρ 1 E r is the regular part of the pencil B + ρ 1 C, which can also be brought into
a canonical form as in (10.20).
Matrix pencils B + ρ 1 C and D + ρ 1 E, of dimension m × n, related by Equation
(10.16), where W and V are square non-singular matrices independent of ρ 1 , are called
strictly equivalent. The eigenvector ṽ in the eigenvalue problem (B + ρ 1 C)ṽ =0is
transformed conformably to w = V −1 ṽ such that the following equation holds:
W −1 (B + ρ 1 C)V (V −1 ṽ)=(D + ρ 1 E)w =0. (10.25)
As shown in Theorem 10.3, the Kronecker Canonical form allows for a decomposition
of a singular matrix pencil into a regular and a singular part. The singular part
relates to all the so-called indeterminate eigenvalues. The regular part is what we are
interested in since it relates to infinite and finite eigenvalues. For our purposes, the
indeterminate, the infinite and the zero eigenvalues will not yield feasible solutions
for our model-order reduction problem and should therefore all be excluded. The
associated Jordan block structure for these eigenvalues can be obtained with exact
arithmetic (if one starts from an exact matrix pencil B +ρ 1 C). In this way it is possible
to arrive algebraically at a smaller sized exact regular matrix pencil in which both
matrices are invertible. Its eigenvalues can then be computed reliably by standard
numerical methods.
In the next subsections the decomposition of a singular matrix pencil into a singular
and a regular part and the deflation of a matrix pencil is discussed. Actually,
the parts corresponding to indeterminate, zero and infinite eigenvalues are split off
using exact arithmetic and the finite eigenvalues of the remaining (regular) part are
computed using numerical arithmetic.