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:本文探讨了迹数最小化方法在两个共轭矩阵对中的适用性。具体来说,我们将回答这个问题:当 $\inf_X\operatorname{tr}(\widehat AX^{\rm H}AX)$ 受制于 $\widehat BX^{\rm H}BX=I$ (大小合适的单位矩阵)时,什么情况下它是有限的?我们获得了充分条件和必要条件,当极值是有限的时候,我们还提出了一个用矩阵对的有限特征值表示的显式公式。我们的结果扩展了 Fan 的迹数最小化原则(1949)和 Kovač-Striko 和 Veselić 对于共轭矩阵对进行的最小化原则(1995),以及我们和合作者们最近对于共轭矩阵对和共轭矩阵进行的最小化原则。