Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem (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 ${\displaystyle Q(\lambda )=\lambda ^{2}M+\lambda C+K}$, with matrix coefficients ${\displaystyle M,\,C,K\in \mathbb {C} ^{n\times n}}$ and we require that ${\displaystyle M\,\neq 0}$, (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.