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10.2. LINEARIZING A POLYNOMIAL EIGENVALUE PROBLEM 167
generalized eigenvalue problem (B + ρ 1 C)w = 0 one may also consider the left generalized
eigenvalue problem u ∗ (B + ρ 1 C)=0 T . Thus, linear dependence of the rows of
B+ρ 1 C for various values of ρ 1 (including infinity) can be given similar meaning too.
As demonstrated in the previous section, the problem of finding solutions of the
system of equations (10.2) subject to (10.3) can be cast in the form of a polynomial
eigenvalue problem involving the square polynomial matrix Ãã N−1
(ρ 1 ) T . The
problem of computing all the values ρ 1 and corresponding non-trivial vectors v that
satisfy Ãã N−1
(ρ 1 ) T v = 0 can be approached by rewriting it as a generalized eigenvalue
problem as follows.
First, expand the polynomial matrix Ãã N−1
(ρ 1 ) T as a matrix polynomial in ρ 1 of
degree N − 1, by:
ÃãN−1 (ρ 1 ) T = A 0 + ρ 1 A 1 + ...+ ρ N−1
1 A N−1 . (10.12)
Applying a linearization technique as mentioned, e.g., in [55], [56] or [72], makes
the polynomial eigenvalue problem (10.10), where the involved matrix is written as
(10.12), equivalent to a generalized eigenvalue problem:
(A 0 + ρ 1 A 1 + ...+ ρ N−1
1 A N−1 ) v =0 ⇐⇒ (B + ρ 1 C)ṽ = 0 (10.13)
where ρ 1 occurs linearly and where the matrices B and C are (large) block-partitioned
square matrices with the following structure:
⎛
⎞
0 I 0
.
B = ⎜
. ..
⎟
⎝ 0 0 I ⎠ ,
A 0 A 1 ... A N−2
⎛
C = ⎜
⎝
(10.14)
⎞
−I 0 0
. .. . ⎟
0 −I 0 ⎠ .
0 ... 0 A N−1
The eigenvectors ṽ of the generalized eigenvalue problem (B+ρ 1 C)ṽ = 0 in (10.13)
have a built-in structure, which is given by:
⎛
ṽ = ⎜
⎝
v
ρ 1 v
.
ρ N−2
1 v
⎞
⎟
⎠ . (10.15)
The block vectors v inside such an eigenvector ṽ are eigenvectors of the original
polynomial eigenvalue problem (10.10). Note that the eigenvectors ṽ have additional