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250 APPENDIX A. LINEARIZING A 2-PARAMETER EIGENVALUE PROBLEM
which is written as:
)
(P 0 + µP 1 + λP 2 w = 0
(A.15)
which is to be enhanced by suitable block rows to fix the particular structure of the
eigenvector w. To achieve this goal, various ways exist. Now the problem is linear in
both µ and λ and the dimensions of the matrices P 0 , P 1 , and P 2 are (m 2 n) × (m 2 n).
Note that the eigenvector w is highly structured, since the vectors from which it is
constructed, the vectors z, are structured too: w =(z T ,λz T ,λ 2 z T ,...,λ m−1 z T ) T
where z =(v T ,µv T ,µ 2 v T ,...,µ m−1 v T ) T .
The eigenvalues of the original problem (A.1) can be found among the eigenvalues
of (A.15) because the eigenvalue spectrum is unchanged. Moreover, the possible
multiplicities among the eigenvalues in (A.1) are preserved when linearizing the polynomial
problem. The eigenvectors of (A.1) can be recovered from the eigenvectors of
(A.15) by making use of the internal structure of the eigenvectors w.
Remark A.1. Note that the matrix M(µ, λ) in (A.1) does not necessarily has to
be square. Suppose that this matrix M is not square but rectangular. Then the
linearization technique of this section can still be applied if the dimensions of the
block matrices I and 0 are adapted conformably.
A.3 Example
Suppose a two-parameter eigenvalue problem is given involving a n × n matrix M
polynomial in µ and λ. The total degree of the entries in the parameters µ and λ in
the matrix M is three: m = 3. We can write the matrix M(µ, λ) as:
M(µ, λ) =N 0 + µN 1 + λN 2 + µ 2 N 3 + µλN 5 +
λ 2 N 4 + µ 3 N 6 + λ 2 µN 7 + λµ 2 N 8 + λ 3 N 9 .
Using this, the polynomial matrix M(µ, λ) can be written as:
M(µ, λ) =
(
(N 0 + λN 2 + λ 2 N 4 + λ 3 N 9 )+
µ (N 1 + λN 5 + λ 2 N 7 )+
µ 2 (N 3 + λN
) 8 )+
µ 3 (N 6 ) .
(A.16)
(A.17)
When the matrices (N 0 + λN 2 + λ 2 N 4 + λ 3 N 9 ), (N 1 + λN 5 + λ 2 N 7 ), (N 3 + λN 8 )
and (N 6 ) are denoted by the matrices M 0 (λ),M 1 (λ),M 2 (λ),M 3 (λ), respectively, the
following formulation is equivalent:
M(µ, λ) =M 0 (λ)+µM 1 (λ)+µ 2 M 2 (λ)+µ 3 M 3 (λ).
(A.18)