On generalizing trace minimization principles

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