Quantum Physics Division

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On linear combinations of two commuting hypergeneralized projectors

O. M. Baksalary^{1}
,
J. Benítez^{2}

^{1} Institute of Physics, Adam Mickiewicz University, ul. Umultowska 85, PL 61-614 Poznań, Poland

^{2} Departamento de Matemática Aplicada, Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia, Camino de Vera s/n. 46022, Valencia, Spain

**Computers and Mathematics with Applications, 56, 2481 (2008)**

Abstract:

The concept of a hypergeneralized projector as a matrix H satisfying H^2=H^@?, where H^@? denotes the Moore-Penrose inverse of H, was introduced by Grosz and Trenkler in [J. Grosz, 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. Grosz, On some linear combinations of hypergeneralized projectors, Linear Algebra Appl. 413 (2006) 264-273] characterized some situations in which a linear combination c"1H"1+c"2H"2, where c"1,c"2@?C and H"1, H"2 are hypergeneralized projectors such that H"1H"2=@h"1H"1^2+@h"2H"2^2=H"2H"1 for some @h"1,@h"2@?C, 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 H"1H"2=H"2H"1.

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