Trace minimization principles for positive semi-definite pencils

This paper is concerned with XHAX) subject to XHBX=J for a Hermitian matrix pencil A-λB, where J is diagonal and J2=I (the identity matrix of apt size). The same problem was investigated earlier by Kovač-Striko and Veselić (Linear Algebra Appl. 216 (1995) 139–158) for the case in which B is assumed nonsingular. But in this paper, B is no longer assumed nonsingular, and in fact A-λB is even allowed to be a singular pencil. It is proved, among others, that the infimum is finite if and only if A-λB is a positive semi-definite pencil (in the sense that there is a real number λ0 such that A-λ0B is positive semi-definite). The infimum, when finite, can be expressed in terms of the finite eigenvalues of A-λB. Sufficient and necessary conditions for the attainability of the infimum are also obtained.