Further results on the reverse order law for \(\{1,3\}\)-inverse and \(\{1,4\}\)-inverse of a matrix product
For a complex \(m \times n\) matrix \(A\), a generalized inverse \(X\) of \(A\) is a matrix, of size \(n \times m\), which satisfies some of the following equations: \[ (1)\;AXA=A, \;\;(2) \;XAX=X, \;\;(3) \;(AX)^*=AX, \;\;(4) \;(XA)^*=XA, \] where \(A^*\) denotes the conjugate transpose of \(A\). Let \(\eta\) be a nonempty subset of \(\{1,2,3,4\}\). In this paper, \(A \eta\) denotes the set of all matrices \(X\) which satisfy (i) for all \(i \in \eta\). Any matrix \(X \in A \eta\) is called an \(\eta\)-inverse of \(A\). One usually denotes any \(\{1\}\)-inverse of \(A\) by \(A^{(1)}\) or \(A^{-}\), and any \(\{1,3\}\)-inverse of \(A\) by \(A^{(1,3)}\) which is also called a least squares \(g\)-inverse of \(A\). Any \(\{1,4\}\)-inverse of \(A\) is denoted by \(A^{(1,4)}\) which is also called a minimum norm \(g\)-inverse of \(A\). The unique \(\{1,2,3,4\}\)-inverse of \(A\) is called the Moore-Penrose generalized inverse of \(A\). Firstly, in this work the authors give the minimal rank of \(D-B^{(1,3)}A^{(1,3)}\) with respect to any \(B^{(1,3)}\) and \(A^{(1,3)}\), being \(A\), \(B\) and \(D\) complex matrices of size \(m \times n\), \(n \times k\) and \(k \times m\), respectively. Secondly, the authors present necessary and sufficient conditions for the inclusion \(B\{1,3\}A\{1,3\}\supseteq (AB)\{1,3\}\). Specifically, they prove that the following statements are equivalent: {\parindent=6mm \begin{itemize}\item[(a)] \(B\{1,3\}A\{1,3\}\supseteq (AB)\{1,3\}\), \item[(b)] \(r(A^*AB,B)+r(A)=r(AB)+\min \{ r(A^*,B), \max \{n+r(A)-m,n+r(B)-k \} \}\), \end{itemize}} where \(r(A)\) denotes the rank of the matrix \(A\). Finally, the authors show necessary and sufficient conditions for \(B\{1,3\}A\{1,3\}=(AB)\{1,3\}\), \(B\{1,4\}A\{1,4\}\supseteq (AB)\{1,4\}\) and \(B\{1,4\}A\{1,4\}= (AB)\{1,4\}\).
scientific article; zbMATH DE number 5823161