Core and strongly core orthogonal matrices

Abstract

In this paper we define two new concepts for a pair of complex square matrices of the index at most 1, the core orthogonal pair and strongly core orthogonal pair. The relationship between core orthogonality (resp. strong core orthogonality) and associated projectors of a pair of matrices is established. Using the core commutativity of two matrices of the index at most 1, a characterization of strongly core orthogonal matrices is also provided. We give several sufficient conditions under which a given pair of matrices is core orthogonal (resp. strongly core orthogonal). A canonical form for a pair of core orthogonal (resp. strong core orthogonal) matrices is developed to obtain characterizations of core orthogonality (resp. strongly core orthogonality). The properties of core orthogonality, strong core orthogonality, core additivity, and rank additivity are studied and their possible interrelationships are discussed. Further, connections of each one of these with the core partial order are also investigated.