Solution to a rank equation

Characterizations of the core inverse and the core partial ordering

“…Analogous results for the Moore-Penrose inverse, group inverse, Drazin inverse and the outer inverse of A with prescribed range and null space can be seen in [5][6][7][8]. Motivated by these relation, in what follows we consider related result for the core inverse of A.…”

“…Groß [5] generalized this rank relation when A is a rectangular matrix. The close relationship between A, B and C is revealed when X = A † is the unique solution as mentioned below.…”

“…Lemma 2.2 [5] Let A ∈ C m,n be of rank r and have representation where U ∈ C m,m and V ∈ C n,n are unitary and D is an r ×r real positive definite diagonal matrix. Then X = A † is the solution to (2.5) if and only if…”