The expression of the generalized Bott-Duffin inverse and its perturbation theory
It is well-known that the Boot-Duffin inverse \(A_L^{-1}=P_L(AP_L+I-P_L)^{-1}\) gives the solution of the constrained system \(Ax+y=b\), where \(x \in L\), \(y \in L^{\perp}\) being \(L\) a subspace of \({\mathbb C}^{n}\), \(L^{\perp}\) its orthogonal, \(A \in {\mathbb C}^{n \times n}\), \(b \in {\mathbb C}^{n}\) and \(P_L\) the projection on \(L\). In this work, an extension of \(A_L^{-1}\) is given by the generalized Boot-Duffin inverse \(A^{(+)}_L = P_L(AP_L+I-P_L)^+\) where \((AP_L+I-P_L)^+\) means the Moore-Penrose inverse of a block of \(AP_L+I-P_L\). Applying this new expression for \(A^{(+)}_L\) a perturbation analysis is developed for the generalized Boot-Duffin inverse.
scientific article; zbMATH DE number 1861222