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

Abstract

An important class of generalized eigenvalue problems Ax=λBx is those in which A and B are Hermitian and some real linear combination of them is definite. For the quadratic eigenvalue problem (QEP) $\text{(}\text{l}\text{̷}{}^2\text{A+}\text{l}\text{̷}\text{B+C)x=0}$ with Hermitian A, B and C and positive definite A, particular interest focuses on problems in which (x^*Bx)²−4(x^*Ax)(x^*Cx) is one-signed for all non-zero x—for the positive sign these problems are called hyperbolic and for the negative sign elliptic. The important class of overdamped problems arising in mechanics is a sub-class of the hyperbolic problems. For each of these classes of generalized and quadratic eigenvalue problems we show how to check that a putative member has the required properties and we derive the distance to the nearest problem outside the class. For definite pairs (A,B) the distance is the Crawford number, and we derive bisection and level set algorithms both for testing its positivity and for computing it. Testing hyperbolicity of a QEP is shown to reduce to testing a related pair for definiteness. The distance to the nearest non-hyperbolic or non-elliptic n×n QEP is shown to be the solution of a global minimization problem with n−1 degrees of freedom. Numerical results are given to illustrate the theory and algorithms.