On Generalizing Trace Minimization

Abstract

Ky Fan's trace minimization principle is extended along the line of the Brockett cost function $\mathrm{trace}(D{X}^{H}AX)$ in $X$ on the Stiefel manifold, where $D$ of an apt size is positive definite. Specifically, we investigate ${\inf }_X\mathrm{trace}(D{X}^{H}AX)$ subject to $X^{H}BX=I_k$ or $J_k=\mathrm{diag}(\pm 1)$. 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 $A-\lambda B$, where $B$ is possibly indefinite and singular, and $D$ is also possibly indefinite.

21 pages

Cited by in corpus (3)

• A Theory of the NEPv Approach for Optimization On the Stiefel Manifold
• Spectrally Constrained Optimization
• On Generalizing Trace Minimization Principles, II