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10.3. THE KRONECKER CANONICAL FORM OF A MATRIX PENCIL 177
Note that the corresponding equations of L T η w 1 = 0 are:
⎧
ρ 1 w 1,1 =0
⎪⎨
w 1,1 + ρ 1 w 1,2 =0
.
w 1,ε−1 + ρ 1 w 1,ε =0
⎪⎩
w 1,ε =0
(10.41)
It is easy to see from (10.41) that only the trivial solution w 1 = 0 exists. The value
of ρ 1 does not play a role in (10.41). The blocks L T η are said to generate indeterminate
eigenvalues ρ 1 too.
The quantities ε i and η i are called the minimal indices for, respectively, the
columns and the rows of a pencil. One property of strictly equivalent pencils is
that they have exactly the same minimal indices.
During the transformation process, some special situations can occur: (i) if ε 1 +
ε 2 +...+ε s = m, then the dimension of the pencil D s +ρ 1 E s is 0×0, (ii) if r = n, then
there are no dependent columns and therefore the blocks L ε will be absent, (iii) if
r = m, then there are no dependent rows and therefore the blocks L T η will be absent.
Finally the pencil B + ρ 1 C is transformed into a quasi block diagonal matrix as
in (10.17) containing a regular (hence square) block D r + ρ 1 E r .
10.3.2 The regular part of a pencil
A regular pencil D r + ρ 1 E r is square, say of dimension n r × n r , such that its determinant
does not vanish identically. In general, if a pencil is regular and the characteristic
polynomial of degree k ≤ n r is equal to p k (ρ 1 )=det(D r + ρ 1 E r ), then the pencil has
k finite eigenvalues (counting multiplicities) which are the zeros of p k (ρ 1 ) and n r − k
infinite eigenvalues.
The regular part D r + ρ 1 E r in (10.17) can be transformed into a Jordan canonical
form as in (10.20), containing the Jordan blocks N 1 ,...,N r1 , J 1 ,...,J r2 and
R 1 ,...,R r3 (see [34] and [42]). These blocks correspond with infinite, zero and nonzero
finite eigenvalues, respectively. The sizes of the blocks N i , J i , and R i , the
associated structure and the eigenvalues are all uniquely determined by the matrices
B and C (but the order in which the blocks appear may vary).
In the application to H 2 model-order reduction we do not compute all these regular
blocks. Once the pencil is brought into the form (10.17), we remove the infinite
and zero eigenvalues of the regular part by exact arithmetic and proceed with the
remaining regular pencil, containing only finite non-zero eigenvalues, using conventional
numerical methods. We will show in this subsection the canonical form of
a regular matrix pencil, containing the Jordan blocks of infinite, zero and non-zero
finite eigenvalues.