Reverse order laws for generalized inverses of multiple matrix products
Let \(A\) be an \(m\times n\) matrix over \(\mathbb{C}\). The Moore-Penrose inverse of \(A\) is an \(n\times m\) matrix \(X\) such that the following four conditions are satisfied: (1) \(AXA=A,\) (2) \(XAX=X\), (3) \(AX\) is Hermitian, and (4) \(XA\) is Hermitian. The Moore-Penrose inverse is unique and is denoted by \(A^{+}\). On the other hand, a generalized inverse satisfying only condition (1) (often called a g-inverse and denoted by \(A^{-}\)) is not usually unique. The author is interested in studying conditions on the \(A_{i}\) under which \( A_{n}^{-}A_{n-1}^{-}\cdots A_{1}^{-}\) is a g-inverse of \(A_{1}\cdots A_{n-1}A_{n}\) for all choices of \(A_{i}^{-}.\) The case for \(n=2\) has previously been studied by \textit{N. Shinozaki} and \textit{M. Sibuya} [Linear Algebra Appl. 9, 29-40 (1974; Zbl 0293.15006)] and the analogous problem for \(A^{+}\) has been considered by \textit{Y. Tian} [Linear Algebra Appl. 211, 85-100 (1994; Zbl 0812.15002)]. The present paper includes a number of necessary and sufficient conditions for the property to hold, most of them expressed in terms of the multiple product-singular value decomposition and quite technical. The least technical of condition to state is: \(A_{n}^{-}A_{n-1}^{-}\cdots A_{1}^{-}\) is a g-inverse of \(A_{1}\cdots A_{n-1}A_{n}\) for all choices of \(A_{i}^{-}\) if and only if either (a) for some \(i\) the range \(\mathcal{R(}A_{i+1}\cdots A_{n})\) is contained in the null space \(\mathcal{N}(A_{1}\cdots A_{i})\) or (b) \( \mathcal{N(}A_{1}\cdots A_{i})\) is contained in \(\mathcal{R}(A_{i+1})\) for each \(i\). The author also proves an analogous theorem for the generalized inverses which satisfy both conditions (1) and (2).
scientific article; zbMATH DE number 1398474