Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem^[1] (QEP), is to find scalar eigenvalues ${\displaystyle \lambda }$, left eigenvectors ${\displaystyle y}$ and right eigenvectors ${\displaystyle x}$ such that

${\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,}$

where ${\displaystyle Q(\lambda )=\lambda ^{2}A_{2}+\lambda A_{1}+A_{0}}$, with matrix coefficients ${\displaystyle A_{2},\,A_{1},A_{0}\in \mathbb {C} ^{n\times n}}$ and we require that ${\displaystyle A_{2}\,\neq 0}$, (so that we have a nonzero leading coefficient). There are ${\displaystyle 2n}$ eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. ${\displaystyle Q(\lambda )}$ is also known as a quadratic polynomial matrix.

Applications

Systems of differential equations

Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:

${\displaystyle Mq''(t)+Cq'(t)+Kq(t)=0}$ 

Where ${\displaystyle q(t)\in \mathbb {R} ^{n}}$ , and ${\displaystyle M,C,K\in \mathbb {R} ^{n\times n}}$ . If all quadratic eigenvalues of ${\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K}$  are distinct, then the solution can be written as

${\displaystyle q(t)=\sum _{j=1}^{2n}\alpha _{j}x_{j}e^{\lambda _{j}t}=Xe^{\Lambda t}\alpha }$ 

Where ${\displaystyle \Lambda ={\text{Diag}}([\lambda _{1},\ldots ,\lambda _{2n}])\in \mathbb {R} ^{2n,2n}}$ , ${\displaystyle X=[x_{1},\ldots ,x_{2n}]\in \mathbb {R} ^{n,2n}}$ , and ${\displaystyle \alpha =[\alpha _{1},\cdots ,\alpha _{2n}]^{\top }\in \mathbb {R} ^{2n}}$ . Stability theory for linear equations can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.

Finite element methods

A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, ${\displaystyle Q(\lambda )}$  has the form ${\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K}$ , where ${\displaystyle M}$  is the mass matrix, ${\displaystyle C}$  is the damping matrix and ${\displaystyle K}$  is the stiffness matrix. Other applications include vibro-acoustics and fluid dynamics.

Methods of solution

Direct methods for solving the standard or generalized eigenvalue problems ${\displaystyle Ax=\lambda x}$  and ${\displaystyle Ax=\lambda Bx}$  are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials. One approach is to transform the quadratic matrix polynomial to a linear matrix pencil (${\displaystyle A-\lambda B}$ ), and solve a generalized eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.

The most common linearization is the first companion linearization

${\displaystyle L(\lambda )={\begin{bmatrix}0&I_{n}\\-K&-C\end{bmatrix}}-\lambda {\begin{bmatrix}I_{n}&0\\0&M\end{bmatrix}},}$ 

where ${\displaystyle I_{n}}$  is the ${\displaystyle n}$ -by-${\displaystyle n}$  identity matrix, with corresponding eigenvector

${\displaystyle z={\begin{bmatrix}x\\\lambda x\end{bmatrix}}.}$ 

We solve ${\displaystyle L(\lambda )z=0}$  for ${\displaystyle \lambda }$  and ${\displaystyle z}$ , for example by computing the Generalized Schur form. We can then take the first ${\displaystyle n}$  components of ${\displaystyle z}$  as the eigenvector ${\displaystyle x}$  of the original quadratic ${\displaystyle Q(\lambda )}$ .

References

1. F. Tisseur and K. Meerbergen, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235–286.