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Eigenvalue problem
In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues
λ
{\displaystyle \lambda }
, left eigenvectors
y
{\displaystyle y}
and right eigenvectors
x
{\displaystyle x}
such that
Q
(
λ
)
x
=
0
and
y
∗
Q
(
λ
)
=
0
,
{\displaystyle Q(\lambda )x=0{\text{ and }}y^{\ast }Q(\lambda )=0,}
where
Q
(
λ
)
=
λ
2
A
2
+
λ
A
1
+
A
0
{\displaystyle Q(\lambda )=\lambda ^{2}A_{2}+\lambda A_{1}+A_{0}}
, with matrix coefficients
A
2
,
A
1
,
A
0
∈
C
n
×
n
{\displaystyle A_{2},\,A_{1},A_{0}\in \mathbb {C} ^{n\times n}}
and we require that
A
2
≠
0
{\displaystyle A_{2}\,\neq 0}
, (so that we have a nonzero leading coefficient). There are
2
n
{\displaystyle 2n}
eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem.
Q
(
λ
)
{\displaystyle Q(\lambda )}
is also known as a quadratic polynomial matrix.
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