Special generalized inverse matrices connected with the theory of unified least squares
Let the symbols \(A^T\), \(A^-\) and \(R(A)\) denote the transpose, any generalized inverse, and the range of \(A\in \mathbb{R}^{m \times n}\) respectively; \((A:B)\) denotes the partitioned matrix composed from the matrices \(A\) and \(B\) that have the same number of rows. In this paper, for a nonnegative definite matrix \(V\) and a matrix \(X\) with the same number of rows, it is demonstrated how to obtain explicit matrices \(G\) such that the range \(R(X:V) =R(V+ XGG^TX^T)\) and \(V(V+ XGG^TX^T)^- V=V\).
scientific article; zbMATH DE number 1076298