Positive semidefinite interval of matrix pencil and its applications to the generalized trust region subproblems

Abstract

We are concerned with finding the set I⪰(A,B) of real values μ such that the matrix pencil A+μB is positive semidefinite. If A,B are not simultaneously diagonalizable via congruence (SDC), I⪰(A,B) either is empty or has only one value μ. When A,B are SDC, I⪰(A,B), if not empty, can be a singleton or an interval. Especially, if I⪰(A,B) is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval I≻(A,B). If A,B are both singular, then even I⪰(A,B) is an interval, its interior may not be I≻(A,B), but A,B are then decomposed as block diagonals of submatrices A1,B1 with B1 nonsingular such that I⪰(A,B)=I⪰(A1,B1). Applying I⪰(A,B), the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size.