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11.2. LINEARIZING THE TWO-PARAMETER EIGENVALUE PROBLEM 197
To find all the solutions of the system of quadratic equations (11.1) subject to
(11.2) one needs to compute:
(i) all the pairs (ρ 1 ,ρ 2 ) such that the matrices Ãã N−1
(ρ 1 ,ρ 2 ) T and Ãã N−2
(ρ 1 ,ρ 2 ) T
simultaneously become singular. Thus, the problem can be written as a polynomial
two-parameter eigenvalue problem involving both the matrices:
⎧
⎨
⎩
ÃãN−1 (ρ 1 ,ρ 2 ) T v =0
ÃãN−2 (ρ 1 ,ρ 2 ) T v =0
(11.3)
Note that here a common eigenvector v is present in both the equations.
(ii) Furthermore, all the corresponding eigenvectors v are required. These eigenvectors
exhibit the Stetter structure and therefore the values of x 1 ,...,x N can be read
off from the eigenvectors. The N-tuples of values x 1 ,...,x N , thus obtained, together
with the corresponding eigenvalues ρ 1 and ρ 2 , constitute all the solutions of the system
of equations (11.1) which simultaneously satisfy the constraints (11.2). From
these solutions approximations G(s) of order N − 3 can be computed, as shown in
Section 11.4.
A problem as in Equation (11.3) is highly structured and because of the known
background of the problem we are allowed to suppose that it will admit a finite number
of solutions. In general however such a problem will have no solutions. It turns out
that techniques from linearizing a polynomial matrix and the Kronecker canonical
form computation of a singular matrix pencil with one parameter, introduced in
Section 10.3, are useful to reliably compute the solutions of this eigenvalue problem
in two parameters. This is the subject of the Sections 11.2 and 11.3.
11.2 Linearizing the two-parameter eigenvalue problem
A polynomial two-parameter eigenvalue problem (11.3), which involves two polynomial
matrices and one common eigenvector, is, to the best of our knowledge, new
in the literature. A solution method does not readily exist so far. Note that the
problem (11.3) has special additional structure: we know from the algebraic techniques,
used to construct the eigenvalue problem, that the matrices Ãã N−1
(ρ 1 ,ρ 2 ) T
and Ãã N−2
(ρ 1 ,ρ 2 ) T commute and that there will be a finite number of solutions. At
first sight, however, without such additional structure, one would generically expect
the problem (11.3) not to have any solutions, as it would be overdetermined: there
are 2 N+1 equations and only 2 N + 1 degrees of freedom (the 2 N − 1 free entries in the
normalized eigenvector v and the 2 parameters ρ 1 and ρ 2 ). Also, when an eigenvector
v and eigenvalues ρ 1 and ρ 2 are found which satisfy Ãã N−1
(ρ 1 ,ρ 2 ) T v =0,itisin
general unlikely that this same solution will also satisfy Ãã N−2
(ρ 1 ,ρ 2 ) T v =0.
A first step to solve the two-parameter polynomial eigenvalue problem (11.3), is to
construct an equivalent eigenvalue problem which is linear in ρ 1 and ρ 2 . This can be