A recurring theorem about pairs of quadratic forms and extensions: a survey

https://apps.dtic.mil/sti/pdfs/ADA048072.pdf

Pertubation bounds for the definite generalized eigenvalue problem

Many applied problems can be posed as a dynamical system defined on a network with attractive and repulsive interactions. Examples include synchronization of nonlinear oscillator networks; the behavior of groups, or cliques, in social networks; and the study of optimal convergence for consensus algorithm. It is important to determine the index of a matrix, i.e., the number of positive and negative eigenvalues, and the dimension of the kernel. In this paper we consider the common examples where the matrix takes the form of a signed graph Laplacian. We show that the there are topological constraints on the index of the Laplacian matrix related to the dimension of a certain homology group. When the homology group is trivial, the index of the operator is determined only by the topology of the network and is independent of the strengths of the interactions. In general, these constraints give bounds on the number of positive and negative eigenvalues, with the dimension of the homology group counting the number ...

Spectral Theory for Dynamics on Graphs Containing Attractive and Repulsive Interactions

This paper gives optimal backward perturbation bounds and the accuracy of approximate solutions for subspaces associated with certain eigenvalue problems such as the eigenvalue problemAx=?x, the generalized eigenvalue problem βAx=?Bx, and the singular value decomposition of a matrixA. This paper also gives residual bounds for certain eigenvalues, generalized eigenvalues and singular values.

Backward perturbation analysis of certain characteristic subspaces

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

The eigenvalue problem $Ax = \lambda Bx$ is considered where A and B are real symmetric matrices. Perturbation bounds are obtained in case the expression $(x^ * Ax)^2 + (x^ * Bx)^2 $ is bounded away from zero. Numerical methods for the solution of the problem are discussed.

https://www.jstor.org/stable/pdfplus/2156448.pdf

C. R. Crawford

Papers

A Stable Generalized Eigenvalue Problem