On generalizing trace minimization principles

Various trace minimization principles have interplayed with numerical computations for the standard eigenvalue and generalized problems in general, as well important applied including linear response problem from electronic structure calculation symplectic of positive definite matrices that play roles classical Hamiltonian dynamics, quantum mechanics, information, among others. In this paper, Ky Fan's principle is extended along line Brockett cost function tr(DXHAX) X on Stiefel manifold, where D an apt size definite. Specifically, we investigate infX⁡tr(DXHAX) subject to XHBX=Ik (the k×k identity matrix) or XHBX=Jk, Jk=diag(±1). We establish conditions under which infimum finite when it finite, analytic solutions are obtained terms eigenvalues eigenvectors matrix pencil A−λB, B possibly indefinite singular, also indefinite.