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) ( l ̷ 2A+ l ̷ 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.