On linear combinations of two commuting hypergeneralized projectors

Abstract

a b s t r a c t The concept of a hypergeneralized projector as a matrix H satisfying H 2 = H Ď , where H Ď denotes the Moore-Penrose inverse of H, 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 c 1 Hinherits 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 1 H 2 = H 2 H 1 .

On linear combinations of generalized involutive matrices

“…Let us remark that a related problem was solved in [3], where it was studied when a linear combination aA + bB is a hypergeneralized projection provided A, B ∈ C n,n are hypergeneralized projections and a, b ∈ C \ {0}, only under the assumption AB = BA. It is worthy that we study more situations besides the commutativity.…”

Revisitation of Generalized and Hypergeneralized Projectors

“…inspired considerations in [2] and [6] leading to a still partial answer to the question of when a linear combination of two hypergeneralized projectors also belongs to the set C HGP n .…”