On Generalizing Trace Minimization

04/01/2021

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by   Xin Liang, et al.

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Ky Fan's trace minimization principle is extended along the line of the Brockett cost function trace(DX^H AX) in X on the Stiefel manifold, where D of an apt size is positive definite. Specifically, we investigate inf_X trace(DX^H AX) subject to X^H BX=I_k or 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-λ B, where B is possibly indefinite and singular, and D is also possibly indefinite.