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198 CHAPTER 11. H 2 MODEL-ORDER REDUCTION FROM ORDER N TO N-3
achieved by applying the linearization technique of Section 10.2 twice on each of the
matrices Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T . Such an approach is given in Appendix
A including an example to demonstrate the procedure.
By applying the technique of Appendix A to linearize (11.3), we arrive at the
equivalent linear two-parameter eigenvalue problem:
⎧
⎨ (B 1 + ρ 1 C 1 + ρ 2 D 1 ) w =0
(11.4)
⎩
(B 2 + ρ 1 C 2 + ρ 2 D 2 ) w =0
where B i , C i , D i (i =1, 2) are large structured matrices of dimensions (N − 1) 2 2 N ×
(N − 1) 2 2 N . The eigenvector w in (11.4) is built up from products of the original
eigenvector v and powers of ρ 1 and ρ 2 .
In the literature on solving two-parameter linear eigenvalue problems (see, e.g.,
[57], [59], [60]) commonly the problem with distinct eigenvectors is addressed:
⎧
⎨ (X 1 + ρ 1 Y 1 + ρ 2 Z 1 )w 1 =0
(11.5)
⎩
(X 2 + ρ 1 Y 2 + ρ 2 Z 2 )w 2 =0
Therefore this theory can not be applied directly to our problem. Moreover and
more importantly, most of this literature deals with iterative solvers to compute only
some eigenvalues, whereas we need all the eigenvalues of the two-parameter eigenvalue
problem (11.3). No literature is available on linear two-parameter eigenvalue problems
involving a common eigenvector w.
11.3 The Kronecker canonical form of a two-parameter matrix pencil
Due to the linearization technique, the first (N − 1) 2 − 1 block rows capture the
structure imposed on the eigenvector w and the last block row represents the equations
of the original non-linear two-parameter eigenvalue problem. Therefore, the first
((N − 1) 2 − 1)2 N equations of (B 1 + ρ 1 C 1 + ρ 2 D 1 )w = 0, are identical to the first
((N − 1) 2 − 1)2 N equations represented by (B 2 + ρ 1 C 2 + ρ 2 D 2 )w =0.
When all the equations represented by (11.4) are written as one matrix-vector
product with the eigenvector w, this yields:
[ ]
B1 + ρ 1 C 1 + ρ 2 D 2
w =0. (11.6)
B 2 + ρ 1 C 2 + ρ 2 D 2
where the involved matrix contains ((N − 1) 2 − 1)2 N duplicate rows. When all
duplicates are removed and only the unique rows are maintained, an equivalent linear
non-square eigenvalue problem arises:
(B + ρ 1 C + ρ 2 D) w = 0 (11.7)