Triple reverse-order law for weighted generalized inverses
\textit{R. E. Hartwig} [Linear Algebra Appl. 76, 241-246 (1986; Zbl 0584.15002)] established the triple reverse-order law for Moore-Penrose inverses involving the computation of some Moore-Penrose inverses. For a given complex matrix \(A\) of size \(m \times n\), the weighted Moore-Penrose is the unique complex matrix \(X\) of size \(n \times m\) satisfying \(AXA=A\), \(XAX=X\) and \(MAX\) and \(NXA\) are Hermitian where \(M\) and \(N\) are Hermitian positive definite. The results found in this work extend the known cases. The problem studied is when the weighted Moore-Penrose of a triple product coincides with the product of the corresponding weighted Moore-Penrose of each one of the matrices in the reverse order. The obtained conditions do not involve any computation of Moore-Penrose inverses but only some ranks of certain matrices. In addition, some special cases are derived.
scientific article; zbMATH DE number 1860991