Pencils of complex and real symmetric and skew matrices

http://www.ma.man.ac.uk/~ftisseur/reports/htv02.pdf

Detecting a definite Hermitian pair and a hyperbolic or elliptic quadratic eigenvalue problem, and associated nearness problems

This paper surveys nearly all of the publications that have appeared in the last twenty years on the theory of and numerical methods for linear pencils. The survey is divided into the following sections: theory of canonical forms for symmetric and Hermitian pencils and the associated problem of simultaneous reduction of pairs of quadratic forms to canonical form; results on perturbation of characteristic values and deflating subspaces; numerical methods. The survey is self-contained in the sense that it includes the necessary information from the elementary theory of pencils and the theory of perturbations for the common algebraic problem Ax=λx.

Matrix pencils: Theory, applications, and numerical methods

Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A + \lambda B + C$ are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given $Q$ has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for $Q$ to be overdamped. For weakly overdamped $Q$ the iteration is shown to be generically linearly convergent with constant at worst $1/2$, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar $\mu<0$ is provided that lies in the gap between the $n$ largest and $n$ smallest eigenvalues of the $n \times n$ quadratic eigenvalue problem (QEP) $Q(\lambda)x = 0$. Once such a $\mu$ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced, using already available Cholesky factorizations, to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic $Q$ so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.

/pdf/detecting-and-solving-hyperbolic-quadratic-eigenvalue-40j69wzp64.pdf

Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems

http://fac.ksu.edu.sa/sites/default/files/classification_paper_2012.pdf.pdf

C.R. Crawford

Papers

Finding a positive definite linear combination of two Hermitian matrices