link to my thesis

Appendix A
A linearization technique for a
two-parameter eigenvalue problem
Let the polynomial eigenvalue problem:
M(µ, λ)v = 0
(A.1)
be given, where the n × n matrix M is polynomial in µ and λ. The total degree of
µ and λ in the entries of M is denoted by m. Using linearization techniques, one
can construct an equivalent, but larger, eigenvalue problem which is linear in both
λ and µ. One linearization technique is discussed in this appendix but various other
approaches are possible. Other techniques to linearize a two parameter polynomial
matrix can be found in the literature, see e.g., [68] and [69].
The linearization approach discussed in this appendix, first linearizes the problem
with respect to µ and, subsequently, with respect to λ.
A.1 Linearization with respect to µ
Let us first expand the polynomial matrix M(µ, λ) in Equation (A.1) as a polynomial
in µ with coefficients that depend on λ:
M(µ, λ) =M 0 (λ)+M 1 (λ) µ + M 2 (λ) µ 2 + ...+ M m (λ) µ m .
(A.2)
Now a linearization technique can be applied with respect to µ only. The eigenvector
z for the eigenvalue problem, which is linear in µ, is chosen as:
⎛ ⎞
v
µv
z =
µ 2 v
. (A.3)
⎜ ⎟
⎝ . ⎠
µ m−1 v
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