Maximal elements under the three partial orders (Q1200551)

Three useful partial orders on a class of \(n\times n\) complex matrices are studied: The star order, minus order and the Loewner or positive semidefinite order. The authors characterize the maximal elements for each of these, when they exist, in a number of classes including the projections, the idempotents, the unitary matrices, and the contractions. In the class of nilpotent matrices they characterize the maximal elements of the star and minus orders but obtain only partial results describing the maximal elements for the positive semidefinite order.