On linear combinations of two commuting hypergeneralized projectors

The concept of a hypergeneralized projector as a matrix satisfying , where denotes the Moore–Penrose inverse of , was introduced by Groß and Trenkler in [J. Groß, G. Trenkler, Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. Generalizing substantially preliminary observations given therein, Baksalary et al. in [J.K. Baksalary, O.M. Baksalary, J. Groß, On some linear combinations of hypergeneralized projectors, Linear Algebra Appl. 413 (2006) 264–273] characterized some situations in which a linear combination , where and , are hypergeneralized projectors such that for some , inherits the hypergenerality property. In the present paper, the problem considered in the latter paper is revisited and solved completely under the essentially weaker assumption that .