On some linear combinations of hypergeneralized projectors (Q819125)

Let \({\mathbb C}_{m,n}\) be the set of \(m \times n\) complex matrices and \(K^{\dagger}\) stand for the Moore-Penrose inverse of \(K\), where \(K\in {\mathbb C}_{m,n}\). Recently, a complete solution to the problem of a linear combination of two different projectors being also a projector was established by \textit{J. K. Baksalary} and \textit{O. M. Baksalary} [Linear Algebra Appl. 321, 3--7 (2000; Zbl 0984.15021)]. Also, an analogous problem for generalized projectors defined as matrices \(G\) satisfying \(G^2=G^*\) instead of projectors was solved. In this paper, a linear combination \(H=c_1H_{1} + c_2H_2\) of hypergeneralized projectors \(H_1\), \(H_2 \in {\mathbb C}^{HGP}_n\), where \(H \in {\mathbb C}^{HGP}_n\), \({\mathbb C}^{HGP}_n = \{K\in {\mathbb C}_{n,n} :K^2 =K^{\dagger} \}\) and \(c_1\), \(c_2 \in {\mathbb C}\), is analyzed. The concept of a hypergeneralized projector was introduced by \textit{J. Groß} and \textit{G. Trenkler} [Linear Algebra Appl. 264, 463--474 (1997; Zbl 0887.15024)] who also provided several properties. Although a complete solution to the problem when the matrix \(H\) is a hypergeneralized projector is unknown, interesting characterizations of the condition \(H^2 =H^{\dagger}\) are obtained by assuming that nonzero \(H_1\), \(H_2\) satisfy a commutativity relation of the form \(H_1H_2 = \eta_1H^2_1 + \eta_2H^2_2 =H_2 H_1\) for some scalars \(\eta_1\), \(\eta_2 \in {\mathbb C}\).