We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques.Expand

We consider polynomial eigenvalue problems P(A,alpha,beta)x=0 in which the matrix polynomial is homogeneous in the eigenvalue (alpha,beta)uC2. In this framework infinite eigenvalues are on the same… Expand

We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory.Expand

The standard way of solving the polynomial eigenvalue problem of degree $m$ in $n\times n$ matrices is to “linearize” to a pencil in vector space $\mathbb{DL}(P)$ of pencils recently identified and studied by Mackey and Mehrmann.Expand

A standard way of treating the polynomial eigenvalue problem $P(\lambda)x = 0$ is to convert it into an equivalent matrix pencil—a process known as linearization.Expand

Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefficient matrices and involve the inverses… Expand

The most widely used approach for solving the polynomial eigenvalue problem $P(\lambda)x = (\sum_{i=0}^m \l^i A_i) x = 0$ in $n\times n$ matrices $A_i$ is to linearize to produce a larger order pencil $L(\lambda X + Y$, whose eigensystem is then found by any method for generalized eigenproblems.Expand

We develop a new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials that is backward-stable for quadratics that are not too heavily damped.Expand