On some linear combinations of hypergeneralized projectors
J. K. Baksalary1 , O. M. Baksalary2 , J. Groß3
1 Faculty of Mathematics, Informatics and Econometrics, Zielona Góra University, ul. Podgórna 50, PL 65-246 Zielona Góra, Poland
2 Institute of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614 Poznań, Poland
3 Department of Statistics, University of Dortmund, Dortmund, Germany
Linear Algebra and Its Applications, 413 264-273 (2006)
The concept of a hypergeneralized projector as a matrix H satisfying H2 = H†, where H† denotes the Moore–Penrose inverse of H, was introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]. In the present paper, the problem of when a linear combination c1H1 + c2H2 of two hypergeneralized projectors H1, H2 is also a hypergeneralized projector is considered. Although, a complete solution to this problem remains unknown, this article provides characterizations of situations in which (c1H1 + c2H2)2 = (c1H1 + c2H2)† derived under certain commutativity property imposed on matrices H1 and H2. The results obtained substantially generalize those given in the above mentioned paper by Groß and Trenkler.