Condition numbers and perturbation of the weighted Moore--Penrose inverse and weighted linear least squares problem.
Given positive definite matrices \(M\) and \(N\) of orders \(m\) and \(n\) respectively, the ``weighted'' Moore-Penrose inverse \(G\) of an \(m\times n\) matrix \(A\) is the unique solution of \(AGA=A\), \(GAG=G\), \((MAG)^* =MAG\), and \((NGA)^* =NGA\). This paper derives formulae for condition numbers of the weighted Moore-Penrose inverse and for similarly weighted condition numbers of singular linear systems \(Ax=b\), and gives a perturbation analysis of the weighted least squares problem in the case in which the perturbed and unperturbed coefficient matrices have the same rank.
scientific article; zbMATH DE number 2008951