scholarly journals On symmetrization and roots of quadratic eigenvalue problems

1972 ◽  
Vol 9 (4) ◽  
pp. 410-422 ◽  
Author(s):  
J Eisenfeld
Keyword(s):  
Eigenvalue Problems ◽  
Quadratic Eigenvalue Problems ◽  
Quadratic Eigenvalue

Gyroscopically Stabilized Systems: A Class Of Quadratic Eigenvalue Problems With Real Spectrum

1991 ◽  
Vol 44 (1) ◽  
pp. 42-53 ◽  
Author(s):  
Lawrence Barkwell ◽  
Peter Lancaster ◽  
Alexander S. Markus
Keyword(s):  
Hilbert Space ◽  
Real Line ◽  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Real Spectrum ◽  
Hyperbolic Operator ◽  
Quadratic Operator ◽  
Distribution Of Eigenvalues ◽  
Quadratic Eigenvalue Problems ◽  
Quadratic Eigenvalue

AbstractEigenvalue problems for selfadjoint quadratic operator polynomials L(λ) = Iλ2 + Bλ+ C on a Hilbert space H are considered where B, C∈ℒ(H), C >0, and |B| ≥ kI + k-l C for some k >0. It is shown that the spectrum of L(λ) is real. The distribution of eigenvalues on the real line and other spectral properties are also discussed. The arguments rely on the well-known theory of (weakly) hyperbolic operator polynomials.

Solving the real eigenvalues of hermitian quadratic eigenvalue problems via bisection

2015 ◽  
Vol 30 ◽  
pp. 721-743 ◽  
Author(s):  
Hao Li ◽  
Yunfeng Cai
Keyword(s):  
Eigenvalue Problems ◽  
Quadratic Eigenvalue Problem ◽  
The Real ◽  
Numerical Tests ◽  
Spring System ◽  
Quadratic Eigenvalue Problems ◽  
The Difference ◽  
Mass Spring ◽  
Quadratic Eigenvalue ◽  
Inertia Index

This paper considers solving the real eigenvalues of the Quadratic Eigenvalue Problem (QEP) Q(\lambda)x =(\lambda^2M+\lambdaC+K)x = 0 in a given interval (a, b), where the coefficient matrices M, C, K are Hermitian and M is nonsingular. First, an inertia theorem for the QEP is proven, which characterizes the difference of inertia index between Hermitian matrices Q(a) and Q(b). Several useful corollaries are then obtained, where it is shown that the number of real eigenvalues of QEP Q(\lambda)x = 0 in the interval (a, b) is no less than the absolute value of the difference of the negative inertia index between Q(a) and Q(b); furthermore, when all real eigenvalues in (a, b) are semi-simple with the same sign characteristic, the inequality becomes an equality. Based on the established theory, the bisection method (with preprocessing) can be used to compute the real eigenvalues of the QEP by computing the inertia indices. Applications to the calculation of the equienergy lines with k.p model, and also a non-overdamped mass-spring system are presented in the numerical tests.

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