Groß [Linear Algebra Appl. 326 (2001) 215] developed characterizations of the minus and star partial orders between the squares of Hermitian nonnegative definite matrices referring to the concept of the space preordering. In the present paper, his results

One of the results of Groß and Trenkler [Linear Algebra Appl. 264 (1997) 463] asserts that a square complex matrix K is a generalized projector if and only if it is (i) quadripotent, (ii) normal, and (iii) partial isometry. The authors supplemented this statement by proving that condition (iii) in the above characterization can actually be deleted. The purpose of the present note is to show that also an alternative reduction of the set of conditions (i)–(iii) is possible as a consequence of establishing the redundancy of (ii) under the presence of (i) and (iii).