Generalization of the class of projectors and linear combination of generalized projectors

GENERALIZATION OF THE CLASS OF PROJECTORS AND LINEAR COMBINATION OF GENERALIZED PROJECTORS SUMMARY Keywords: Idempotent matrix; Quadripotent matrix; EP matrix; Partial isometry; Projector; Orthogonal projector; Generalized projector; Hypergeneralized projector. This work, first chapter being preliminaries for the other chapters, consists of three principal chapters. In the Chapter 1, some definitions and some theorems that will be fundamental in the Chapter 2 and Chapter 3 are given. In the Chapter 2, generalizations of the class of projectors are considered within the classes of normal and EP matrices. A necessary and sufficient condition for the sum and difference of two generalized projectors to be a generalized projector is given. Moreover, a sufficient condition for the sum and difference of two hypergeneralized projectors to be a hypergeneralized projector is given. Furthermore, a sufficient condition for the product of any two matrices belonging to any one of the classes of idempotents, projectors, generalized projectors, hypergeneralized projectors to be in the same class is given. In the Chapter 3, G1? G2 being any two different nonzero nxn generalized projectors and cx, c2 being nonzero complex numbers, the problem of characterising all situations, where a linear combination of the form G=c1G1+c2G2, is also a generalized projector which is the generalization of the problem concerning the sum Gj + G2, and the difference Gj - G2 in Chapter 2 has been considered. VIII