It is known that within the set of orthogonal projectors (Hermitian idempotent matrices) certain matrix partial orderings coincide in the sense that when two orthogonal projectors are ordered with respect to one of the orderings, then they are also ordered with respect to the others. This concerns, inter alia, the star, minus, diamond, sharp, core, and Löwner orderings. The situation changes, though, when instead of two orthogonal projectors, various functions of the pair (being either no longer Hermitian or no longer idempotent) are compared. The present paper provides an extensive investigation of the matrix partial orderings of functions of two orthogonal projectors. In addition to the six orderings mentioned above, three further binary functions are covered by the analysis, one of which is the space preordering. A particular attention is paid to the requirements that either product, sum, or difference of two orthogonal projectors is itself an orthogonal projector, i.e., inherits both features, Hermitianness and idempotency. Links of the results obtained with the research areas of applied origin (e.g., physics and statistics) are pointed out as well.

众所周知，在正交投影仪（厄米幂等矩阵）的集合中，某些矩阵偏序是一致的，因为当两个正交投影仪相对于其中一个排序时，它们也相对于其他排序进行排序。这尤其涉及星形、减号、菱形、锐利、核心和 Löwner 排序。但是，当不是两个正交投影仪，而是比较这对函数的各种函数（不再是 Hermitian 或不再是幂等的）时，情况就会发生变化。本文对两个正交投影仪的函数的矩阵偏序进行了广泛的研究。除了上面提到的六个排序之外，分析还涵盖了另外三个二元函数，其中之一是空间预排序。特别注意两个正交投影仪的乘积、和或差本身就是正交投影仪的要求，即继承两个特征，厄密性和幂等性。还指出了所获得的结果与应用起源的研究领域（例如物理学和统计学）之间的联系。