On the projectors FF† and F†F

Abstract

Abstract A particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors, namely FF † and F † F , determined by a complex square matrix F and its Moore–Penrose inverse F † . Various functions of the projectors are considered from the point of view of their nonsingularity, idempotency, nilpotency, or their relation to the known classes of matrices, such as EP, bi-EP, GP, DR, or SR. This part of the paper was inspired by Benitez and Rakocevic [J. Benitez, V. Rakocevic, Matrices A such that AA † − A † A are nonsingular, Appl. Math. Comput. 217 (2010) 3493–3503]. Further characteristics of FF † and F † F , with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues, are derived as well. Besides establishing selected exemplary results dealing with FF † and F † F , the paper develops a general approach whose applicability extends far beyond the characteristics provided therein.