Quadratic eigenvalue problem

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the quadratic eigenvalue problem (QEP), is to find <a href="/facts/Scalar_(mathematics)/MkDbA4Eo">scalar</a> <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> 
 
 
 
 λ
 
 
 {\displaystyle \lambda }
 
, left <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a> 
 
 
 
 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
 
 
 M
 +
 λ
 C
 +
 K
 
 
 {\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K}
 
, with matrix coefficients 
 
 
 
 M
 ,
 
 C
 ,
 K
 ∈
 
 
 C
 
 
 n
 ×
 n
 
 
 
 
 {\displaystyle M,\,C,K\in \mathbb {C} ^{n\times n}}
 
 and we require that 
 
 
 
 M
 
 ≠
 0
 
 
 {\displaystyle M\,\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 <a href="/facts/Nonlinear_eigenproblem/nKtKhK4I">nonlinear eigenproblem</a>. 
 
 
 
 Q
 (
 λ
 )
 
 
 {\displaystyle Q(\lambda )}
 
 is also known as a quadratic <a href="/facts/Polynomial_matrix/XTFENTxj">polynomial matrix</a>.

Quadratic eigenvalue problem open-in-new

Quadratic eigenvalue problem