Matrix analysis and applied linear algebra
Matrix analysis and applied linear algebra
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For an nxn complex matrix A and the nxn identity matrix I"n, the difference I"n-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 I"n-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 I"n-A. Several such representations are provided and a number of relevant conclusions originating from them are identified.