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

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A framework for analyzing nonlinear eigenproblems and parametrized linear systems (2011, unknown): “It yields 2/3 ≤ max(‖Ã2‖2, ‖Ã1‖2, ‖Ã0‖2) ≤ 2 and ρP̃ (θopt) = max ( 1, ‖Ã1‖2/(‖Ã0‖2‖Ã2‖2) ) , and so ρ P̃ (θopt) ≈ 1 if the quadratic is elliptic [20], [26] or, in the terminology of quadratics arising from mechanical systems with damping, the system is not too heavily damped.”
Smith Forms of Palindromic Matrix Polynomials (2011, unknown): “Complex ∗-palindromic polynomials occur when the Crawford number of two Hermitian matrices is computed [10], as well as in the solution of discrete-time linear-quadratic optimal control problems via structured eigenvalue problems [2].”
The Quadratic Eigenvalue Problem (2001, background): “New questions arise, including, “Where is the nearest nondefinite or nonoverdamped system?” [75] and “Where is the nearest nonregular quadratic matrix polynomial?” As we have stressed throughout this survey, QEPs are of growing importance and many open questions are associated with them.”
Rayleigh Quotient Based Optimization Methods For Eigenvalue Problems (2014, background): “It was argued in [20] that the hyperbolic quadratic eigenvalue problem (HQEP) is the closest analogue of the standard Hermitian eigenvalue problem Hx = λx when it comes to the quadratic eigenvalue problem (λ(2)A+ λB + C)x = 0.”