The paper concerns definite generalized eigenvalue problems and hyperbolic quadratic eigenvalue problems (QEPs) having the property that all eigenvalues are real. It also concerns elliptic QEPs which have eigenvalues that are all non-real. There are two aims: to determine for a given generalized eigenvalue problem or QEP, whether the property of interest holds, and, if it does, to compute the distance to the nearest problem without that property.
The second section, showing that the distance from a definite generalized eigenvalue problem to the nearest non-definite one is given by the Crawford number \[ \gamma(A,B)= \min_{\substack{ z\in \mathbb{C}^n\\ \|z\|_2=1}} \sqrt{(z^*Az)^2+ (z^*Bz)^2}, \] \(A,B\in \mathbb{C}^{n\times n}\) two Hermitian matrices, contains two methods for the computation of that number and for testing the definiteness. The first one, based on the bisection algorithm, detects definiteness and produces a bracket for \(\gamma(A,B)\), that shrinks to zero. The second one, more efficient and rapidly convergent is obtained by applying the level set algorithm derived for stability radii computations. The procedure enables us to test for definiteness and produces only a monotonically lower bound.
The third section contains definitions and characterizations of hyperbolic (including the subclass of overdamped QEPs) and elliptic QEPs. It shows that testing for hyperbolicity can be reduced to testing for definiteness of an associated generalized eigenvalue problem, which can be done using one of the algorithms presented above. It is proved that the distance from a hyperbolic (elliptic) problem to the nearest non-hyperbolic (non-elliptic) one, given by \[ \begin{split} d(A,B,C)= \min\{f(\Delta A,\Delta B,\Delta C): \text{det}(W(x, A+\Delta A, B+\Delta B,C+\Delta C))= 0,\\ \text{for some }x\neq 0\},\end{split} \] where \(f\) is some nonnegative function of the perturbation matrices \(\Delta A\), \(\Delta B\), \(\Delta C\), can be expressed in terms of a global minimization problem over the unit ball.
Finally, to illustrate the results, the paper presents three numerical examples each with a different type of QEPs: a damped mass-spring system which is real and overdamped, a moving wirsaw which is complex and hyperbolic but not overdamped and a wave equation which is real and elliptic.