On disjoint range matrices

For a square complex matrix F and for F∗ being its conjugate transpose, the class of matrices satisfying R(F)∩R(F∗)={0}, where R(.) denotes range (column space) of a matrix argument, is investigated. Besides identifying a number of its properties, several functions of F, such as F+F∗, (F:F∗), FF∗+F∗F, and F-F∗, are considered. Particular attention is paid to the Moore–Penrose inverses of those functions and projectors attributed to them. It is shown that some results scattered in the literature, whose complexity practically prevents them from being used to deal with real problems, can be replaced with much simpler expressions when the ranges of F and F∗ are disjoint. Furthermore, as a by-product of the derived formulae, one obtains a variety of relevant facts concerning, for instance, rank and range.