On Generalizing Trace Minimization

TL;DR

This paper extends Ky Fan's trace minimization principle to a broader class of matrix optimization problems on the Stiefel manifold, providing conditions for finiteness and explicit solutions involving eigenvalues and eigenvectors.

Contribution

It generalizes trace minimization to cases with indefinite matrices and derives explicit solutions using eigenstructure analysis.

Findings

Conditions for finiteness of the trace minimization problem

Explicit solutions in terms of eigenvalues and eigenvectors

Extension to indefinite and singular matrices

Abstract

Ky Fan's trace minimization principle is extended along the line of the Brockett cost function $\mathrm{trace}(DX^H AX)$ in $X$ on the Stiefel manifold, where $D$ of an apt size is positive definite. Specifically, we investigate $\inf_X \mathrm{trace}(DX^H AX)$ 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.