## 29 Citations

On Generalizing Trace Minimization

- MathematicsArXiv
- 2021

Ky Fan’s trace minimization principle is extended along the line of the Brockett cost function tr(DXAX) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we…

Matrix pencils and existence conditions for quadratic programming with a sign-indefinite quadratic equality constraint

- MathematicsJ. Glob. Optim.
- 2009

A complete characterization of the solution set to the constrained minimization problem in terms of the eigenspace of the matrix pencil is provided.

Linear pencils and quadratic programming problems with a quadratic constraint

- Mathematics
- 2022

Given bounded selfadjoint operators A and B acting on a Hilbert space H, consider the linear pencil P (λ) = A+ λB, λ ∈ R. The set of parameters λ such that P (λ) is a positive (semi)definite operator…

Extensions of Wielandt’s min–max principles for positive semi-definite pencils

- Mathematics
- 2014

There are numerous min–max principles about the eigenvalues of a Hermitian matrix. The most general ones are Wielandt’s min–max principles which include the Courant–Fischer min–max principles and the…

An indefinite variant of LOBPCG for definite matrix pencils

- Mathematics, Computer ScienceNumerical Algorithms
- 2013

The authors' new method can be seen as a variant of the popular LOBPCG method operating in an indefinite inner product and also turns out to be a generalization of the recently proposed LOBP4DCG method by Bai and Li for solving product eigenvalue problems.

Quadratic (Weakly) Hyperbolic Matrix Polynomials: Direct and Inverse Spectral Problems

- Mathematics
- 2009

Let L be a monic quadratic weakly hyperbolic or hyperbolic n × n matrix polynomial. We solve some direct spectral problems: We prove that the eigenvalues of a compression of L to an (n −…

Symplectic eigenvalue problem via trace minimization and Riemannian optimization

- Mathematics
- 2021

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson’s theorem. It is…

Preconditioned gradient iterations for the eigenproblem of definite matrix pairs

- Computer ScienceETNA - Electronic Transactions on Numerical Analysis
- 2019

A unifying framework of preconditioned gradient iterations for definite generalized eigenvalue problems with indefinite B, which compute a few eigenvalues closest to the definiteness interval and the corresponding eigenvectors of definite matrix pairs (A,B), that is, pairs having a positive definite linear combination.

Convergence Analysis of Extended LOBPCG for Computing Extreme Eigenvalues

- Mathematics, Computer ScienceArXiv
- 2020

The algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate.

## References

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A Jacobi eigenreduction algorithm for definite matrix pairs

- Mathematics
- 1993

SummaryWe propose a Jacobi eigenreduction algorithm for symmetric definite matrix pairsA, J of small to medium-size real symmetric matrices withJ2=I,J diagonal (neitherJ norA itself need be…

Trace Minimization Algorithm for the Generalized Eigenvalue Problem

- Computer SciencePPSC
- 1987

An inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace, and the rate of convergence of the method is established, and a technique for improving it is discussed.

The Symmetric Eigenvalue Problem

- Art
- 1980

According to Parlett, 'Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for…