This paper is concerned with establishing a trace minimization principle for two Hermitian matrix pairs. Specifically, we will answer the question: when is $\inf_X\operatorname{tr}(\widehat AX^{\rm H}AX)$ subject to $\widehat BX^{\rm H}BX=I$ (the identity matrix of apt size) finite? Sufficient and necessary conditions are obtained and, when the infimum is finite, an explicit formula for it is presented in terms of the finite eigenvalues of the matrix pairs. Our results extend Fan's trace minimization principle (1949) for a Hermitian matrix, a minimization principle of Kova\v{c}-Striko and Veseli\'c (1995) for a Hermitian matrix pair, and most recent ones by the authors and their collaborators for a Hermitian matrix pair and a Hermitian matrix.

翻译：本文旨在建立两个厄米矩阵对的迹最小化原理。具体地，我们将回答：当$\widehat BX^{\rm H}BX=I$（适当大小的单位矩阵）时，$\inf_X\operatorname{tr}(\widehat AX^{\rm H}AX)$何时是有限的？我们给出了充分必要条件，并且当下确界有限时，我们利用矩阵对的有限特征值给出了其显式表达式。我们的结果推广了厄米矩阵的Fan迹最小化原理（1949）、Kovač-Striko和Veselić（1995）关于厄米矩阵对的最小化原理，以及作者及其合作者最近关于厄米矩阵对和厄米矩阵的结果。