How to characterize equalities for the Moore-Penrose inverse of a matrix (Q2755632)

In the first part of the paper, some results on rank equalities involving a matrix \(A\), the Moore-Penrose inverse \(A^{+}\) and its conjugate transpose \(A^{*}\) are established. Then, these equalities are used to characterize different equalities with \(A\) and \(A^{+}\) . In particular, the rank of the matrix \(AA^{+} - A^{+}A \) is given and from that result, the equality \(AA ^{+}\) = \(A^{+}A\) is characterized, which is a property of range-Hermitian matrices, that is, \(\text{range} (A) = \text{range} (A^{\ast})\). Some other commutative characterizations are given. As the author notes, no matrix decomposition is needed for those characterizations since the proofs are based on rank properties. In the last section, more results on matrix rank expressions involving powers of the Moore-Penrose inverse of a matrix are given.