link to my thesis

168 CHAPTER 10. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-2
structure, because the vectors v are structured too as being Stetter vectors (with a
structure corresponding to the chosen monomial basis).
The matrices B and C in this set-up have dimensions (N − 1) 2 N × (N − 1) 2 N .
This gives rise to (N − 1) 2 N generalized eigenvalues. Because spurious eigenvalues
can be introduced by making the matrix AãN−1 (ρ 1 ) T polynomial in ρ 1 , one has to
check which eigenvalues (which can be numerical values, indeterminate or infinite) are
eigenvalues of the problem (10.10) too. Recall that the roots −1/δ i for all i =1,...,N
of the terms (1+ρ 1 δ 1 ), (1+ρ 1 δ 2 ),...,(1+ρ 1 δ N ) are the quantities that may be introduced
as spurious eigenvalues when constructing the polynomial matrix Ãã N−1
(ρ 1 ) T .
The equivalence between the polynomial eigenvalue problem and the generalized
eigenvalue problem, is expressed by the fact that if one has computed a solution of the
generalized eigenvalue problem (10.13), then this yields a solution of the polynomial
eigenvalue problem (10.10) too, and vice versa. Once the eigenvalues ρ 1 and eigenvectors
ṽ of generalized eigenvalue problem (10.13) are found and the appropriate
ones have been selected, the values x 1 ,...,x N can be read off from the vectors ṽ.
This provides a method to find all the solutions of the polynomial eigenvalue problem
(10.10) and, therefore, to find all the solutions of the system of equations (10.2) that
satisfy the constraint ã N−1 = 0 in (10.3).
10.3 The Kronecker canonical form of a matrix pencil
A matrix pencil B + ρ 1 C of dimensions m × n is regular when the involved matrices
are square (m = n) and the determinant is not identically zero. When a pencil is
singular, then m n, or m = n and the determinant of the matrix is ≡ 0. Therefore,
when a pencil is rectangular it is singular by definition.
The matrix pencil B +ρ 1 C of size (N −1) 2 N ×(N −1) 2 N in (10.13) of the previous
section will in general be singular, although it involves square matrices. Singular
generalized eigenvalue problems may have indeterminate, infinite, zero and non-zero
finite eigenvalues and are well-known to be numerically highly ill-conditioned. The
eigenvalues which cause this singularity of the eigenvalue problem are unwanted as
they do not have any meaning for our H 2 model-order reduction problem. A solution
here is to decompose, using exact arithmetic, the matrix pencil B + ρ 1 C into a singular
part, which corresponds to indeterminate eigenvalues, and a regular part, which
corresponds to infinite and finite eigenvalues, and to split off the unwanted eigenvalues.
This can be achieved by using the computation of the Kronecker canonical form
(see [6], [34] and [42]).
In the conventional square case, the matrix pencil A − ρ 1 I n can be conveniently
characterized by means of its Jordan canonical form. The sizes of the Jordan blocks
for the various eigenvalues of the matrix A provide geometrical information on the
associated eigenspaces, both for right eigenvectors and for left eigenvectors. The Kronecker
canonical form is a generalization of the Jordan canonical form to the possible
rectangular and singular case, and applies to arbitrary m×n matrix pencils B +ρ 1 C.