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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