Sequential determination of the \(\{1, 4\}\)-inverse of a matrix
The authors provide a sequential approach of the \(\{1,4\}\)-generalized inverse of the matrix \(B=[A,a]\), where \(A\) is an \(m\) by \(r\) matrix and \(a\) is an \(m\) by \(p\) block that is appended to it. For that, firstly they give the \(\{1,4\}\)-generalized inverse in terms of the \(\{1,2,4\}\)-generalized inverse of the matrix \(A\). Secondly, they establish the relationship between the \(\{1,4\}\)-generalized inverse of any given matrix and its \(\{1,2,4\}\)-generalized inverse. Finally they show a sequential determination of the \(\{1,4\}\)-generalized inverse of an augmented matrix \(B\) in terms of a \(\{1,4\}\)-generalized inverse of the original matrix \(A\). Moreover, it is proved that if an \(m\) by \(n\) matrix \(A\) has rank \(m\), then its \(\{1,2,4\}\)-generalized inverse is equal to its \(\{1,2,3,4\}\)-generalized inverse. In general, the proofs are rather technical. The authors use the fact that the \(\{1,4\}\)-inverse provides the unique minimum length solution \(x=B^{\{1,4\}}b\), and the Moore-Penrose properties of generalized inverses.
scientific article; zbMATH DE number 1997745