On the matrix difference I−A

For an n×nn×n complex matrix A and the n×nn×n identity matrix In, the difference In−Ais investigated. By exploiting a partitioned representation, several features of such a difference are identified. In particular, expressions for its Moore–Penrose inverse in some specific situations are established, and representations of the pertinent projectors are derived. Special attention is paid to the problem, how certain properties of A and In−A are related. The properties in question deal with known classes of matrices, such as GP, EP, partial isometries, bi-EP, normal, projectors, and nilpotent. An important part of the paper is devoted to demonstrating how to obtain representations of orthogonal projectors onto various subspaces determined by A and/or In−A. Several such representations are provided and a number of relevant conclusions originating from them are identified.