# What is Eigenvalue problem: Definition and 85 Discussions

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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1. G

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9. M

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45. M

### Generalized Eigenvalue problem

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