On the projectors FF† and F†F

摘要

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 Benítez and Rakočević [J. Benítez, V. Rakočević, 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.