Further properties of generalized and hypergeneralized projectors
It is well known that a complex \(n\times n\) matrix \(K\) is a projector on the vector space \({\mathbb C}^{n}\) precisely when \(K^2\) equals the identity matrix \(I_ n\). Moreover, a projector is orthogonal (in terms of the standard unitary scalar product) if, and only if, \(K\) coincides with its conjugate transpose \(K^*\). Following \textit{J.~Groß} and \textit{G.~Trenkler} [Linear Algebra Appl. 264, 463-474 (1997; Zbl 0887.15024)], the authors consider \textit{generalized} and \textit{hypergeneralized} projectors; they are defined via the conditions \(K^ 2= K^ *\) and \(K^ 2 = K^ +\) (the Moore-Penrose inverse of \(K\)), respectively. Several properties of such matrices are established. Some of the results obtained extend and/or generalize those given in the reference above, whereas others are solutions to new problems. For example, the following holds: \(K\) is a (hyper)generalized projector if, and only if, \(K^ +\) shares the same property.
scientific article; zbMATH DE number 2117299