The reverse order law for the generalized inverse \(A^{(2)}_{T,S}\)
Let \(A\) be a matrix \(m \times n\) over the complex number field \(\mathbb{C}\), with \(rank\) \(A=n\), and let \(T\) and \(S\) be subspaces of \(\mathbb{C}^{n}\) and \(\mathbb{C}^{m}\) respectively, with dim \(T=\) dim \(S^{\perp }=t\leq r\). Then \(A\) has a \(\{2\}\)-inverse \(A_{T,S}^{(2)}\) such that \(range\) \(A_{T,S}^{(2)}=T\) and \(null space\) \(A_{T,S}^{(2)}=S\) if and only if \(AT \oplus S= \mathbb{C} ^{m}\) [cf. \textit{A. Ben-Israel} and \textit{T. N. E. Greville}, Generalized inverses: Theory and applications (Wiley, New York) (1974; Zbl 0305.15001)]. The reverse order law for the generalized inverse of the multiple matrix products yields a class of interesting problems that have attracted attention since 1966, when \textit{T. N. E. Greville} [SIAM Rev. 8, 518--521 (1966; Zbl 0143.26303)] gave a necessary and sufficient condition for reverse order law for the Moore-Penrose inverse, \((AB)^{\dagger }=B^{\dagger }A^{\dagger }\). In this paper the authors present a necessary and sufficient condition by a rank identity for the reverse order law \( (A_{1}A_{2} \cdots A_{n})_{T,S}^{(2)}=(A_{n})_{T_{n},S_{n}}^{(2)} \cdots (A_{2})_{T_{2},S_{2}}^{(2)}(A_{1})_{T_{1},S_{1}}^{(2)} \). As special cases, similar results are also derived for the Moore-Penrose inverse \(A^{\dagger }\), the weighted Moore-Penrose inverse \(A_{M,N}^{\dagger }\), the Drazin inverse \(A_{d}\) and the group inverse \(A_{g}\), respectively, of \(A\).
scientific article; zbMATH DE number 2107889
scientific article