A partial ordering approach to characterize properties of a pair of orthogonal projectors

Authors:

• Oskar Maria Baksalary,
• Götz Trenkler

Abstract

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.